The visitors had taken the lead in the first half when Willian showed great control inside the area to slot in.
It was no less than Fulham deserved after an impressive first half, but Liverpool stepped it up after the break and fought back with two goals in three minutes.
First, Curtis Jones equalised when his 68th-minute strike from distance took a big deflection off Tosin Adarabioyo to wrong-foot Bernd Leno in the Fulham goal.
Substitute Cody Gakpo then drove in a Darwin Nunez pull-back to send the home crowd wild.
It was the second time in just over a month that Liverpool had denied Fulham success at Anfield. Marco Silva's side had led 3-2 in the final 10 minutes of their Premier League clash on 3 December but lost 4-3.
The two sides will meet again in the second leg at Craven Cottage on Wednesday, 24 January, with Jurgen Klopp's men now in the driving seat as they look to keep alive their hopes of a record-extending 10th League Cup win.
'These are the moments we should learn from'
FT: Liverpool 2-1 Fulham
Fulham
Fulham manager Marco Silva speaking to BBC Radio 5 Live: "Yes [very disappointing and tough to take]. A very good first half, we controlled it really well. We didn't give them any space, no space in behind. We know that when you control our back line really well then they have problems.
"We could have in some moments got some more goals as we would have liked but it went to plan. We were ruthless, we scored the first chance that we got, and after they didn't have many [chances] to be honest. One from Elliott which was offside but nothing else - no dangerous moments that I can pick out from memory.
"Second half the substitutions with Darwin and Gakpo, they had more threat in attack, more pace, the quality to create problems - created much more for us to defend.
"Even so I believe we had a good 10-15 minutes or so with very good chances to score the 2-0 and to create a much better scenario for us. We had the moments and with better finishing and a bit more belief we could have at least scored again.
"We didn't and of course lose a challenge in the middle of the pitch and their first goal is a bit lucky - a deflection on a shot from the edge of the box. They are the moments we should learn from, with this crowd, at these stadiums, against these teams, for the next 5-10 minutes you have to be really smart."
'In the second half it was a proper, proper Liverpool performance'
FT: Liverpool 2-1 Fulham
Liverpool
Liverpool manager Jurgen Klopp, speaking to Sky Sports: "It was super tough. First half we caused ourselves some problems. They were really compact defending and we had to change at half-time and the boys did really well.
"They had big counter-attacking chances, then we were dominant, we scored and everything was fine. It's not the best game we've ever played but we're human beings.
"It was always too slow and we had to change it. It was then a good game and the crowd were there all the time for us, we just didn't give them a lot to cheer about. But in the second half it was a proper, proper Liverpool performance.
"Fulham are a really good team. Can you play pretty offensively against us? No, then we hit you on the counter attack. It will be an open game and we want to go to the final and if we do a really good game against Fulham and a good result we will. If not, congratulations to Fulham and they go to the final."
On the performance of Conor Bradley, Klopp added: "Exceptional, what a boy, what fantastic character and the potential. He is in the right team, everyone loves him and respects him and wants him to succeed."
'He has this natural scouse confidence, a joy to have around'
FT: Liverpool 2-1 Fulham
Liverpool
Getty ImagesCopyright: Getty Images
Liverpool boss Jurgen Klopp speaking about Curtis Jones to BBC Radio 5 Live: "Curtis [Jones] is super, super, super important to us and he knows that. Coincidentally I heard his interview and the only thing to break through was 'I need more game time'. *laughs* I love that! So sorry Curtis that I didn't give you more game time earlier.
"He has made big steps. He looked like the fittest player on the whole pitch. He got a knock, I think a dead leg, he went down, but then a second later he turned around and scored.
"A good footballer, very, very good - he always was... but using these skills, he is now for me, for him for everybody thank god, doing it consistently. That is really cool. He made big steps in defensive things, which gives him a better feeling for the game. Then all of a sudden he can show his quality more.
"It is a joy to have the boy around because he has this natural scouse confidence and it is really cool."
Comeback kings
FT: Liverpool 2-1 Fulham
Tonight against Fulham was the seventh time Liverpool have won a match despite conceding the first goal in all competitions this season, the most of any side in England’s top four tiers in 2023-24.
'We believe and we will fight for it until the very end'
FT: Liverpool 2-1 Fulham
Fulham
Fulham manager Marco Silva, speaking to BBC Radio 5 Live: "You cannot concede another goal, you have to manage the game. A fast counter attack away from home, drawing in a competition where you have the second game at home, you cannot concede.
"The reality is that they score in that moment. But OK, they are winning 2-1 at half time, we have a game to play at home and we are for sure going to fight for it. We are going to do our best to play in the final."
On his message to the players ahead of the second leg: "Of course that is the message [Fulham can take it to this Liverpool side]. I told the players that straight away. We believe in ourselves, and we will fight for it until the very end. It is an ambition for us. It is going to be difficult for us, but it is going to be difficult for Liverpool too.
"We play at home, it is important for our crowd to create the right environment, to get behind us and to do their part, to help us to beat Liverpool and to reach the final."
'We're going to go for it for sure'
FT: Liverpool 2-1 Fulham
Fulham
Fulham manager Marco Silva, speaking to Sky Sports, said: "We knew it was going to be a tough one. You can't play for a final thinking it's not going to be really hard. Even three days ago they were able to play Arsenal away from home in the FA Cup.
"We did really well in the first half, more of a low block than we usually play but we controlled it well. They are a threat with the pace and quality of their attacking players and they can punish you at any time. We were ruthless in the first half but second half they pushed up a bit more.
"They started to create more problems, we had 10-15 minutes of good chances and De Cordova-Reid was a moment for us to score. We didn't take that moment and we lost a few challenges we shouldn't have done and the shot from the edge of the box took a deflection.
"We have to be more clever from a tactical point of view and we should defend our box better for their second goal.
"We're playing at home, we're losing 2-1 but we're going to go for it for sure. In the second half we had good moments and we should've scored another goal at least but it is what it is at the top level of football.
"At home, with our fans behind us, it's going to be a great night for Fulham Football Club."
'Plenty of changes to the team but we keep getting the job done'
FT: Liverpool 2-1 Fulham
Liverpool
Man of the Match Curtis Jones, speaking to BBC Radio 5 Live: "It was an enjoyable evening. The first half was a tough game, we had most of the ball, we dominated, but we couldn't score. In the second half we turned that around, we scored two good goals and dominated again. So good game all round."
On if it has been a special week in the cups: "Yeah definitely. It is one of those things that there are plenty of games coming round, we are making changes to the team but we are still getting the job done. So happy days."
On getting a good run in the team: "It is benefitting me loads. The important thing is that I keep it up. My chance is there, so it is on me now [to perform]."
On the second leg: "It is always a tough task until the end in all games. We're going there with a lead but the tie isn't over, so we'll see on the day."
'We showed good mentality'
FT: Liverpool 2-1 Fulham
Liverpool
Goalscorer Cody Gakpo, speaking to Sky Sports, said: "He [Curtis Jones] got a very important goal to get us back in the game and after that I felt we really dominated, not only with possession but creating chances and then the second so I was happy.
"It's the mentality in the group, in the club, in the city, everyone is like this mentality of just keep going and it doesn't matter what happens and we showed that again today. If you have that mentality a lot of things can happen.
"I love playing with these guys. We showed good mentality and have to keep going as it's only the first leg of this game."
'Fulham were made to pay for missed chances'
FT: Liverpool 2-1 Fulham
Mark Schwarzer
Former Fulham keeper on BBC Radio 5 Live
ReutersCopyright: Reuters
Fulham will be very, very disappointed with the outcome of the game.
There was a 20-minute period in the second half where they had a couple of really big chances and could have, or really should have, taken advantage and made it two nil.
I was telling myself 'they are going to be made to pay for this,' particularly the chance for Bobby De Cordova-Reid. And they were made to pay.
Liverpool looked like they were out of ideas, and like they were a bit all over the place. The substitutions made all the difference. Who's to say they wouldn't have done that anyway and brought them back in the game [despite the chances].
On the flip side it was a good second half from Liverpool and could have been even better if it weren't for the Bernd Leno save from Darwin Nunez. That would have been game over.
'Leaving with smiles on their faces'
FT: Liverpool 2-1 Fulham
Liverpool
Goalscorer Curtis Jones, speaking to Sky Sports, said: "In the first half it was tough, we had all the ball but couldn't find the back of the net and in the second half we changed that around. The fans are leaving with smiles on their faces so I'm happy.
"I came all the way through the academy so it [scoring] means more. I've been around the team since I was 17 and the journey has been up and down.
"It has not always been easy but I've always stuck in there, my mindset is right, I know I have the ability and I just need the game time."
Making an impact
FT: Liverpool 2-1 Fulham
Despite only coming on in the 56th minute, Darwin Nunez had the highest expected goals (0.76), the most touches in the opposition box (12), completed the most dribbles (3) and had the joint most shots (4) of any Liverpool player tonight, as well as assisting both goals.
Liverpool's home run continues
FT: Liverpool 2-1 Fulham
Gary Rose
BBC Sport at Anfield
Getty ImagesCopyright: Getty Images
The home fans are going away happy as their side extend their incredible home record this season to 14 wins and two draws.
A victory looked unlikely for the Reds after a first half when Fulham had more chances while also defending strongly.
Liverpool were a different prospect after the break and in the end fully deserved the win.
Cody Gakpo and Darwin Nunez had the required impact from off the bench, but the win could have looked more comfortable had Nunez scored one of his chances.
There's just one goal in it and Fulham will fancy their chances of putting up a fight on home soil in the second leg.
'Second leg will be a completely different game'
FT: Liverpool 2-1 Fulham
Dion Dublin
Former England striker on BBC Radio 5 Live
I think the tie just leans in Liverpool's favour a little bit as they now know how good Fulham can be in fits and starts and what they can do on the break.
Fulham won't play the same at Craven Cottage, they will be more positive, want more of the ball and Liverpool will have to defend a little bit more. It will be a completely different game in the second leg.
The tie is in the balance but I would still fancy Liverpool.
Post update
FT: Liverpool 2-1 Fulham
Cody Gakpo has scored in all four of his League Cup games for Liverpool; he is the first player to score in four successive appearances in the competition for the club since Vladimir Smicer in 2000-01.
BBC SportCopyright: BBC Sport
'Liverpool went through the gears and took over'
FT: Liverpool 2-1 Fulham
Dion Dublin
Former England striker on BBC Radio 5 Live
Liverpool went through the gears in the second half, changed the game completely. The introductions of Darwin Nunez and Cody Gakpo at half time allowed Diogo Jota to go on and do what he does best, working from a little bit deeper in midfield.
Nunez and Gakpo did really well up top, combined for one of the goals.
Fulham played their part in this game. Led for 40 minutes, and had their chances to make it 2-0 too but didn't.
Liverpool bided their time, got the fans involved in the game and took over.
Post update
FT: Liverpool 2-1 Fulham.
So it's all to do for both Chelsea and Fulham, although both sides will have home advantage for the second legs, with Chelsea facing Championship Middlesbrough on 23 January, before Fulham host Liverpool a day later.
Get Involved
#bbcfootball, text 81111 (UK only - standard rates apply) or WhatsApp 03301231826
SMS Message: Great game. Fulham a bit naive, though. You don't keep going punch for punch at Anfield when Liverpool have just equalised, and are throwing everything forward. Looking forward to the 2nd leg. from NMP
Great game. Fulham a bit naive, though. You don't keep going punch for punch at Anfield when Liverpool have just equalised, and are throwing everything forward. Looking forward to the 2nd leg.
SMS Message: Liverpool equivalent of 2-1 up at half time. Won't be used to that. Fulham still in with a good shout from Dom, Surrey
Liverpool equivalent of 2-1 up at half time. Won't be used to that. Fulham still in with a good shout
SMS Message: There are a lot of Liverpool fans getting carried away. We're at home and have been average at best against a limited and uninspired Fulham team. from Geoff
There are a lot of Liverpool fans getting carried away. We're at home and have been average at best against a limited and uninspired Fulham team.
FULL-TIME
Liverpool 2-1 Fulham
Getty ImagesCopyright: Getty Images
That's all for today. Liverpool, just as Middlesbrough did yesterday, gained a one-goal advantage before a second-leg trip to west London.
Fulham took the lead, but will be relieved to still be in the cup tie after the Reds missed a number of chances late on with Bernd Leno doing superbly to keep them down to two.
The second leg is at Craven Cottage in two weeks' time.
Live Reporting
Michael Emons
All times stated are UK
Getty ImagesCopyright: Getty Images 
Getty ImagesCopyright: Getty Images 
ReutersCopyright: Reuters 
Getty ImagesCopyright: Getty Images 
BBC SportCopyright: BBC Sport 
Getty ImagesCopyright: Getty Images
Latest PostThanks and goodbye
FT: Liverpool 2-1 Fulham
That's all for this live text.
Thanks for joining us and for your comments. Until next time, bye for now.
Liverpool fight back to beat Fulham in EFL Cup - read the report
FT: Liverpool 2-1 Fulham
Gary Rose
BBC Sport at Anfield
Liverpool staged a second-half turnaround to beat Fulham at Anfield in the first leg of their Carabao Cup semi-final tie.
The visitors had taken the lead in the first half when Willian showed great control inside the area to slot in.
It was no less than Fulham deserved after an impressive first half, but Liverpool stepped it up after the break and fought back with two goals in three minutes.
First, Curtis Jones equalised when his 68th-minute strike from distance took a big deflection off Tosin Adarabioyo to wrong-foot Bernd Leno in the Fulham goal.
Substitute Cody Gakpo then drove in a Darwin Nunez pull-back to send the home crowd wild.
It was the second time in just over a month that Liverpool had denied Fulham success at Anfield. Marco Silva's side had led 3-2 in the final 10 minutes of their Premier League clash on 3 December but lost 4-3.
The two sides will meet again in the second leg at Craven Cottage on Wednesday, 24 January, with Jurgen Klopp's men now in the driving seat as they look to keep alive their hopes of a record-extending 10th League Cup win.
'These are the moments we should learn from'
FT: Liverpool 2-1 Fulham
Fulham
Fulham manager Marco Silva speaking to BBC Radio 5 Live: "Yes [very disappointing and tough to take]. A very good first half, we controlled it really well. We didn't give them any space, no space in behind. We know that when you control our back line really well then they have problems.
"We could have in some moments got some more goals as we would have liked but it went to plan. We were ruthless, we scored the first chance that we got, and after they didn't have many [chances] to be honest. One from Elliott which was offside but nothing else - no dangerous moments that I can pick out from memory.
"Second half the substitutions with Darwin and Gakpo, they had more threat in attack, more pace, the quality to create problems - created much more for us to defend.
"Even so I believe we had a good 10-15 minutes or so with very good chances to score the 2-0 and to create a much better scenario for us. We had the moments and with better finishing and a bit more belief we could have at least scored again.
"We didn't and of course lose a challenge in the middle of the pitch and their first goal is a bit lucky - a deflection on a shot from the edge of the box. They are the moments we should learn from, with this crowd, at these stadiums, against these teams, for the next 5-10 minutes you have to be really smart."
'In the second half it was a proper, proper Liverpool performance'
FT: Liverpool 2-1 Fulham
Liverpool
Liverpool manager Jurgen Klopp, speaking to Sky Sports: "It was super tough. First half we caused ourselves some problems. They were really compact defending and we had to change at half-time and the boys did really well.
"They had big counter-attacking chances, then we were dominant, we scored and everything was fine. It's not the best game we've ever played but we're human beings.
"It was always too slow and we had to change it. It was then a good game and the crowd were there all the time for us, we just didn't give them a lot to cheer about. But in the second half it was a proper, proper Liverpool performance.
"Fulham are a really good team. Can you play pretty offensively against us? No, then we hit you on the counter attack. It will be an open game and we want to go to the final and if we do a really good game against Fulham and a good result we will. If not, congratulations to Fulham and they go to the final."
On the performance of Conor Bradley, Klopp added: "Exceptional, what a boy, what fantastic character and the potential. He is in the right team, everyone loves him and respects him and wants him to succeed."
'He has this natural scouse confidence, a joy to have around'
FT: Liverpool 2-1 Fulham
Liverpool
Liverpool boss Jurgen Klopp speaking about Curtis Jones to BBC Radio 5 Live: "Curtis [Jones] is super, super, super important to us and he knows that. Coincidentally I heard his interview and the only thing to break through was 'I need more game time'. *laughs* I love that! So sorry Curtis that I didn't give you more game time earlier.
"He has made big steps. He looked like the fittest player on the whole pitch. He got a knock, I think a dead leg, he went down, but then a second later he turned around and scored.
"A good footballer, very, very good - he always was... but using these skills, he is now for me, for him for everybody thank god, doing it consistently. That is really cool. He made big steps in defensive things, which gives him a better feeling for the game. Then all of a sudden he can show his quality more.
"It is a joy to have the boy around because he has this natural scouse confidence and it is really cool."
Comeback kings
FT: Liverpool 2-1 Fulham
Tonight against Fulham was the seventh time Liverpool have won a match despite conceding the first goal in all competitions this season, the most of any side in England’s top four tiers in 2023-24.
'We believe and we will fight for it until the very end'
FT: Liverpool 2-1 Fulham
Fulham
Fulham manager Marco Silva, speaking to BBC Radio 5 Live: "You cannot concede another goal, you have to manage the game. A fast counter attack away from home, drawing in a competition where you have the second game at home, you cannot concede.
"The reality is that they score in that moment. But OK, they are winning 2-1 at half time, we have a game to play at home and we are for sure going to fight for it. We are going to do our best to play in the final."
On his message to the players ahead of the second leg: "Of course that is the message [Fulham can take it to this Liverpool side]. I told the players that straight away. We believe in ourselves, and we will fight for it until the very end. It is an ambition for us. It is going to be difficult for us, but it is going to be difficult for Liverpool too.
"We play at home, it is important for our crowd to create the right environment, to get behind us and to do their part, to help us to beat Liverpool and to reach the final."
'We're going to go for it for sure'
FT: Liverpool 2-1 Fulham
Fulham
Fulham manager Marco Silva, speaking to Sky Sports, said: "We knew it was going to be a tough one. You can't play for a final thinking it's not going to be really hard. Even three days ago they were able to play Arsenal away from home in the FA Cup.
"We did really well in the first half, more of a low block than we usually play but we controlled it well. They are a threat with the pace and quality of their attacking players and they can punish you at any time. We were ruthless in the first half but second half they pushed up a bit more.
"They started to create more problems, we had 10-15 minutes of good chances and De Cordova-Reid was a moment for us to score. We didn't take that moment and we lost a few challenges we shouldn't have done and the shot from the edge of the box took a deflection.
"We have to be more clever from a tactical point of view and we should defend our box better for their second goal.
"We're playing at home, we're losing 2-1 but we're going to go for it for sure. In the second half we had good moments and we should've scored another goal at least but it is what it is at the top level of football.
"At home, with our fans behind us, it's going to be a great night for Fulham Football Club."
'Plenty of changes to the team but we keep getting the job done'
FT: Liverpool 2-1 Fulham
Liverpool
Man of the Match Curtis Jones, speaking to BBC Radio 5 Live: "It was an enjoyable evening. The first half was a tough game, we had most of the ball, we dominated, but we couldn't score. In the second half we turned that around, we scored two good goals and dominated again. So good game all round."
On if it has been a special week in the cups: "Yeah definitely. It is one of those things that there are plenty of games coming round, we are making changes to the team but we are still getting the job done. So happy days."
On getting a good run in the team: "It is benefitting me loads. The important thing is that I keep it up. My chance is there, so it is on me now [to perform]."
On the second leg: "It is always a tough task until the end in all games. We're going there with a lead but the tie isn't over, so we'll see on the day."
'We showed good mentality'
FT: Liverpool 2-1 Fulham
Liverpool
Goalscorer Cody Gakpo, speaking to Sky Sports, said: "He [Curtis Jones] got a very important goal to get us back in the game and after that I felt we really dominated, not only with possession but creating chances and then the second so I was happy.
"It's the mentality in the group, in the club, in the city, everyone is like this mentality of just keep going and it doesn't matter what happens and we showed that again today. If you have that mentality a lot of things can happen.
"I love playing with these guys. We showed good mentality and have to keep going as it's only the first leg of this game."
'Fulham were made to pay for missed chances'
FT: Liverpool 2-1 Fulham
Mark Schwarzer
Former Fulham keeper on BBC Radio 5 Live
Fulham will be very, very disappointed with the outcome of the game.
There was a 20-minute period in the second half where they had a couple of really big chances and could have, or really should have, taken advantage and made it two nil.
I was telling myself 'they are going to be made to pay for this,' particularly the chance for Bobby De Cordova-Reid. And they were made to pay.
Liverpool looked like they were out of ideas, and like they were a bit all over the place. The substitutions made all the difference. Who's to say they wouldn't have done that anyway and brought them back in the game [despite the chances].
On the flip side it was a good second half from Liverpool and could have been even better if it weren't for the Bernd Leno save from Darwin Nunez. That would have been game over.
'Leaving with smiles on their faces'
FT: Liverpool 2-1 Fulham
Liverpool
Goalscorer Curtis Jones, speaking to Sky Sports, said: "In the first half it was tough, we had all the ball but couldn't find the back of the net and in the second half we changed that around. The fans are leaving with smiles on their faces so I'm happy.
"I came all the way through the academy so it [scoring] means more. I've been around the team since I was 17 and the journey has been up and down.
"It has not always been easy but I've always stuck in there, my mindset is right, I know I have the ability and I just need the game time."
Making an impact
FT: Liverpool 2-1 Fulham
Despite only coming on in the 56th minute, Darwin Nunez had the highest expected goals (0.76), the most touches in the opposition box (12), completed the most dribbles (3) and had the joint most shots (4) of any Liverpool player tonight, as well as assisting both goals.
Liverpool's home run continues
FT: Liverpool 2-1 Fulham
Gary Rose
BBC Sport at Anfield
The home fans are going away happy as their side extend their incredible home record this season to 14 wins and two draws.
A victory looked unlikely for the Reds after a first half when Fulham had more chances while also defending strongly.
Liverpool were a different prospect after the break and in the end fully deserved the win.
Cody Gakpo and Darwin Nunez had the required impact from off the bench, but the win could have looked more comfortable had Nunez scored one of his chances.
There's just one goal in it and Fulham will fancy their chances of putting up a fight on home soil in the second leg.
'Second leg will be a completely different game'
FT: Liverpool 2-1 Fulham
Dion Dublin
Former England striker on BBC Radio 5 Live
I think the tie just leans in Liverpool's favour a little bit as they now know how good Fulham can be in fits and starts and what they can do on the break.
Fulham won't play the same at Craven Cottage, they will be more positive, want more of the ball and Liverpool will have to defend a little bit more. It will be a completely different game in the second leg.
The tie is in the balance but I would still fancy Liverpool.
Post update
FT: Liverpool 2-1 Fulham
Cody Gakpo has scored in all four of his League Cup games for Liverpool; he is the first player to score in four successive appearances in the competition for the club since Vladimir Smicer in 2000-01.
'Liverpool went through the gears and took over'
FT: Liverpool 2-1 Fulham
Dion Dublin
Former England striker on BBC Radio 5 Live
Liverpool went through the gears in the second half, changed the game completely. The introductions of Darwin Nunez and Cody Gakpo at half time allowed Diogo Jota to go on and do what he does best, working from a little bit deeper in midfield.
Nunez and Gakpo did really well up top, combined for one of the goals.
Fulham played their part in this game. Led for 40 minutes, and had their chances to make it 2-0 too but didn't.
Liverpool bided their time, got the fans involved in the game and took over.
Post update
FT: Liverpool 2-1 Fulham.
So it's all to do for both Chelsea and Fulham, although both sides will have home advantage for the second legs, with Chelsea facing Championship Middlesbrough on 23 January, before Fulham host Liverpool a day later.
Get Involved
#bbcfootball, text 81111 (UK only - standard rates apply) or WhatsApp 03301231826
FULL-TIME
Liverpool 2-1 Fulham
That's all for today. Liverpool, just as Middlesbrough did yesterday, gained a one-goal advantage before a second-leg trip to west London.
Fulham took the lead, but will be relieved to still be in the cup tie after the Reds missed a number of chances late on with Bernd Leno doing superbly to keep them down to two.
The second leg is at Craven Cottage in two weeks' time.