It took Brendan Rodgers' side 60 minutes, and three changes, to get going at Parkhead as Alistair Johnston made it two goals from two games by tapping home Yang Hyun-Jun's cross.
Arne Engels made it two seven minutes later when he calmly converted from the spot after Kyogo Furuhashi won a penalty.
And the league leaders saw it out from there against a Dundee side who have now just won twice in 10 and slip to 10th place.
Image source, SNSDundee manager Tony Docherty told BBC Scotland: "I was really pleased with the level of performance the players put in. Playing against a top, top opponent like Celtic there's so many things to get right.
"Coming in at nothing each [at half-time] I thought 'more of the same' and we did that.
"You saw the quality of substitutes Celtic brought on and I thought we were unlucky to lose the first goal.
"I felt a little bit hard done by for the second goal, I don't think it was a penalty. I thought the performance of the players merited a clean sheet.
"I'm just proud of the players tonight, the mentality and the work rate was really pleasing.
"The most pleasing aspect tonight, we had three academy graduates in the middle of the pitch and I thought they equipped themselves well. For a club like Dundee that's what we're all about, improving and developing players in that arena. That's the most pleasing thing, as well as our team performance.
"I demand levels of performance and I thought tonight the level was really strong. If we continue to show that spirit, mentality and quality on the ball then we'll be ok."
Charlotte Cohen
BBC Sport Scotland
Image source, SNSAfter the weekend's disappointment of dropping three points from a winning position, there will be similar feelings in the Dundee dressing room after that first half at Parkhead.
Dundee held off Brendan Rodgers' side for 60 minutes, forcing the Celtic manager into a triple change shortly after the break to switch things up which ultimately made the difference.
But there was a feeling in the first 45 minutes that maybe Celtic wouldn’t be able to break down the robust Dundee backline.
They did though, twice, and a solid first half display doesn't change that Tony Docherty's side head back to Dens Park with nothing.
Two wins in 10 games, 10th place in the table and things don't get any easier - a visit from Killie on Sunday before a trip to the fortress of Pittodrie the following weekend awaits.
Brendan Rodgers has rung the changes once again, making six alterations to the side that started the 3-0 victory over Motherwell.
Alistair Johnston, Liam Scales, Alex Valle, Reo Hatate, James Forrest and Daizen Maeda all drop to the bench. There's no Callum McGregor in the squad tonight.
Cameron Carter-Vickers is back in the starting line-up after making his return off the bench at the weekend, Greg Taylor is also back in at left-back while Anthony Ralston replaces Johnston on the right.
Arne Engels, Hyun-Jun Yang, and Luis Palma are also in from the start.
Worth noting that it's a first league start of the season for Palma, Yang and Ralston.
Tony Docherty has made three changes from the 2-1 defeat to St Johnstone. Trevor Carson is replaced by Jon McCracken between the sticks due to a knock. Josh Mulligan and Antonio Portales are in for Curtis Main and Billy Koumetio.
Image source, SNSCeltic are unbeaten in 39 league meetings with Dundee (W33 D6); only against Partick Thistle have they ever gone 40+ league matches in a row without defeat against a single opponent (48 from 1897 to 1923).
Dundee have lost 19 of their last 21 away games against Celtic in all competitions (D2) since a 2-0 victory in May 2001.
Celtic are unbeaten in 37 midweek (Tuesday, Wednesday, Thursday) home games in the Scottish Premiership (W33 D4), winning their last 22 in a row by an aggregate score of 63-6. The Hoops’ last such defeat came to St Johnstone in March 2015 (1-0).
Dundee are winless in their last 40 league meetings with reigning Scottish champions (D6 L34) since a 2-0 win away to Rangers in March 2001 under Ivano Bonetti.
Brendan Rodgers has won 12 of his 13 meetings with Dundee as Celtic manager in all competitions, drawing the other 0-0 in April 2018.
Tyrone Smith
BBC Sport Scotland Senior Reporter
Dundee have not won back-to-back league games since March – a statistic that tells its own story.
Consistency of results has become an issue, and it is something manager Tony Docherty will need to find a way of addressing if his side are to try to better last season’s impressive sixth-place Premiership finish.
It is an easy thing to say, but a harder thing to fix.
Docherty must be looking at the impact his former talisman Luke McCowan is now starting to make at Celtic and dearly wish he was still able to call on his services.
The Dark Blues have won only one game since McCowan departed.
While one player doesn't make a team, the midfielder's creativity – and goals – are clearly a huge miss to the Dens Park side.
Dundee are likely to come up against McCowan when they travel to Celtic on league business tonight.
Not the fixture you would chose if you were looking for a match that could potentially serve as a catalyst to get things moving in the right direction once again.
But sometimes when little is expected, teams can deliver. Docherty will be hoping that is the case in Glasgow's east end this evening.
Image source, SNSCeltic striker Adam Idah will miss out with an ankle injury while Callum McGregor remains a doubt with a hip strain. Greg Taylor could make his comeback from a calf problem.
Dundee goalkeeper Trevor Carson will drop out with a knock and Scott Fraser remains unavailable but Jordan McGhee is fit again. Joe Shaughnessy is working his way back from a long-term knee injury.
Image source, SNSMcCowan and Docherty in August, prior to the latter's move to Celtic
Dundee manager Tony Docherty says he is "not surpirsed at all" with how well former captain Luke McCowan is doing for Celtic.
McCowan made his first start for Celtic on Sunday and scored the opening goal in a 3-0 win at Motherwell, just four days after helping the Hoops to a Champions League point against Atalanta in Italy.
The 26-year-old midfielder has scored two goals in seven appearances in all and has impressed with his ability to find space and his use of the ball.
McCowan comes up against Dundee on Wednesday night at Parkhead, two months after moving from Dens Park.
Docherty said: "I'm not surprised at all to see how well he's doing at Celtic because I knew the player that we had here, what a talented boy he was.
"Not just a talented footballer but a good person. And he really responded when I came into the club in terms of giving a wee bit more leadership and authority.
"But he has now moved on and he is an opponent and we will be doing all we can to make sure he is not a happy boy [on Wednesday] night".
Ethan Hampton
Fan writer
Saturday's match was, unfortunately, the poorest performance I've seen from Tony Docherty's Dundee since we were mauled 7-1 at Parkhead last season.
Initially starting well, Simon Murray's goal was a slice of individual brilliance as he lifted the ball over the goalkeeper and allowed it to drop from the heavens before sending the home crowd to cloud nine.
We almost doubled our lead immediately, the tails were up and Ziyad Larkeche forced Ross Sinclair into a big save. At that point we looked like we could cruise to three points - but then we never looked like scoring again.
Dundee just surrendered possession, it was like a home game for St Johnstone, and any team that attempts to absorb pressure for a substantial amount of time with no intent of laying another finger on the opposition deserve to concede goals.
That's not the way Dundee should play at Dens Park and I can only assume it was a deliberate tactical decision.
Trevor Carson did not deserve to be on the losing team, he produced two top-drawer saves and even got credited with the assist for Murray’s goal. He deserved more from his team-mates.
Throwing away a lead and losing to a 90th-minute goal feels like classic Dundee.
The victory at Fir Park should have been a turning point, a showcase of expelling the demons that have plagued us for so long, but instead it was a step forward just to slip and fall back on our bottoms.
With a trip to Celtic on Wednesday we are bracing ourselves to be kicked while we're down. We haven't beaten the Hoops since 2001 and I am very doubtful that will change any time soon.
It's likely to be damage control until we host Kilmarnock on Saturday and that's when the pressure will begin to amplify. Just three points above the relegation zone and three straight home defeats is unacceptable and such a disheartening course of events given how well we started the season.
I hope to see a defiant reaction this week. Dundee were great for bouncing back from negative results last season and the fans must see that level of resilience again.
Dundee will discover next week if their proposed new training ground at Riverside Drive will get the go ahead when councillors consider the planning application submitted by Dundee FC Community Trust. (The Courier), external
Watch highlights as St Johnstone beat Dundee 2-1 in the Scottish Premiership.
Sportscene pundits Neil McCann and Lee Miller reflect on Dundee's poor form in their 2-1 defeat against St Johnstone.
Image source, SNSTony Docherty says his Dundee side must learn to be less vulnerable from set pieces after Saturday's defeat to St Johnstone.
Simon Murray's goal put the hosts in front after 12 minutes before Graham Carey equalised with half an hour to play at Dens Park.
But Nicky Clark's 90th minute goal secured all three points for Simo Valakari's side.
"I should be in there speaking to the players and bemoaning the fact we didn’t control the game like we can," Docherty said.
"But we got the rewards for going two up against their central defenders.
"They had a lot of the ball but they never hurt us and we went ahead.
"I wanted more control but it was heading for a 1-1 and then we could have picked the bones out of it.
"To then lose it from a set-play at the end, amid all that confusion, is bitterly disappointing.
"It’s frustrating, especially after we kept a clean sheet at a difficult venue last week.
"But we made attacking changes and I always want to be brave and hope to win the game.
"To lose it from a set-play is something we need to go through and learn from."
Dundee fans, we asked for your views on Saturday's 2-1 defeat to St Johnstone.
Here's what some of you said:
Doug: Same old excuses Tony Docherty, getting a bit fed up now maybe it's time to think about your position if this isn't resolved. This has been going on since last season and isn't getting any better. We should be in a top six position but instead we are four points off the bottom, not good enough.
Brian: Disgraceful performance, totally outplayed and outfought, the midfield were brushed off the ball too easily and the two forwards were given little to play with. Trevor Carson made some cracking saves. Man of the match for me goes to big Billy Koumetio, he was outstanding. Docherty, you need to have a long hard look to see if you are up for the job as we appear to be floundering badly.
Jim: We need to stop dropping points from winning positions, lost count of how many times that has happened the past two seasons. We do have a really young squad right enough, we missed some experience against St Johnstone I think.
Danny: We are now in big trouble and it's all down to the manager's continuing inability to make the correct substitutions at the right time. He has received deserved praise for some of our attacking play but his win ratio overall is poor.
Gavin: It seems difficult to see what's changed from last season. From a cohesive team, to one that looks like confidence is as low as it can get. We have to turn the corner quickly or it’s going to be a relegation battle at Dens this season unfortunately!
Andy: An awful performance. The scoreline flattered Dundee who deserved to lose by two or three. Second best all over the park. Tactics were baffling; why spend the second half trying to absorb pressure and hang on to a one goal lead with a defence notorious for their individual errors and lapses of concentration? Surely more sensible to attack and finish the game off?
Doug: An insipid performance by Dundee gained them exactly what they deserved; nothing. They were devoid of attacking flair for the majority of the game and far too many times they decided to play the ball backwards instead of putting the St Johnstone defence under pressure. Lyall Cameron was particularly guilty in this regard. The better team won.
Image source, SNSDundee have lost a home game after leading at half time in the Premiership for the first time since 5th October 2024 against Kilmarnock .
St Johnstone have won after trailing at half time away from home in the Premiership for the first time since 17th December 2022 against Ross County (14 games without a win after trailing at half time).
Simon Murray has scored three goals in the Premiership this season (one in this game), more than any other Dundee player.
St Johnstone have attempted 20 shots in this game, their highest total in a single match in the Premiership this season.
St Johnstone battled back to earn a Scottish Premiership victory against Dundee after a dramatic finish to Simo Valakari's first game in charge.
What did you make of Dundee's display? And should Nicky Clark's late goal have stood?
Tony Docherty discusses 2-1 defeat to St Johnstone
Dundee manager Tony Docherty: "He [referee Kevin Clancy] explained to us the length of time it took. He said there was a handball involved in the build up. They couldn't make a decision whether it was offside or not by drawing the lines.
"The explanation was there was so many bodies in there they couldn't determine whether Nicky Clark was offside. I don't understand that and why that amount of time wasn't added on at the end of the game. I'm a bit confused.
"It comes down to the 89th minute and not defending a corner properly. Again we've let ourselves down there. We need to address that.
"These times are vital in games and we need someone to take responsibility for that. We didn't do that.
"We did everything like that last week absolutely right because we worked really hard on it, now it's happened again.
"It's a young group and a new group so it's important we're honest and identify the responsibility for it. We'll sort that out and make sure it's right for the next game."
Thomas Duncan
BBC Sport Scotland
Image source, SNSDundee dug out a big win on the road against Motherwell last time, keeping their first clean sheet of the season.
It felt like an opportunity to build on that against a St Johnstone side they have a good record against, and on their own patch.
But, other than Simon Murray's industriousness creating the goal, they were strangely limp in front of their own fans.
Substitutes Scott Tiffoney and Oluwaseun Adewumi injected some life, but not enough to prevent a painful defeat.
Tony Docherty's side have not won back-to-back league matches since March, and the Dundee boss will be frustrated they can't get going in the way they did in the early part of last season.
They have played well in a lot of games this season without getting a victory, but this was not one of them.
Image source, SNSSimon Murray came up with the brightest moment for Dundee on a tough afternoon.
Image source, SNSMo Sylla is back after his red card against Kilmarnock
Dundee midfielder Mo Sylla returns from suspension, while defender Jordan McGhee and midfielder Scott Fraser will be assessed.
And centre-half Joe Shaughnessy is making good progress from a long-term knee injury.
Croat centre-half Bozo Mikulic is still waiting on a work permit to complete his move to St Johnstone after leaving Partizan Tirana.
The visitors will also again be without Cammy MacPherson, Uche Ikpeazu and Sam McClelland.
Image source, SNSDundee midfielder Mo Sylla returns from suspension while Jordan McGhee and Scott Fraser will be assessed. Joe Shaughnessy is making good progress from a long-term knee injury.
St Johnstone will again be without Cammy MacPherson, Uche Ikpeazu (knee) and Sam McClelland (Achilles), while Croatian centre-half Bozo Mikulic is still waiting on a work permit.
Image source, SNSRyan Astley wants to kick on after the defender marked his first league start for Dundee with a win and clean sheet.
The 23-year-old signed in January and featured in the League Cup but had only featured once as a late substitute in the Premiership before getting in the team for last weekend's win at Motherwell.
Astley is hoping to keep his place when St Johnstone visit Dens on Saturday.
"It's been difficult," he said. "We have a really good squad, I knew that coming up here, and that I would have to fight to get a chance in the team.
"I feel like I've done that, I've never sulked, just got on with it and worked hard, and hopefully I have taken my chance.
"I just want to kick on, and show to the fans firstly what I can do, what I have come here for, to the gaffer, and just enjoy my football.
"I really enjoyed the weekend, obviously the result helped, but I'm ready to kick on now and show what I can do.
"Obviously I want to play every game, who wouldn't want to? But as a team my main focus is on winning games and keeping clean sheets, and that keeps me in the team."
Image source, SNSTony Docherty insists his Dundee side "owe" the home supporters a win against St Johnstone on Saturday.
While the Dark Blues ended a six-game winless run by triumphing at Motherwell last weekend, they have lost their last two home outings, including squandering a 2-0 lead in a late collapse against Kilmarnock.
"With that result [at Fir Park] we've given ourselves a platform that hopefully we can go and build on," said manager Docherty.
"It's good to be back at home, I said at Motherwell I wanted to reward the away travelling support because they’ve been fantastic.
"The home support has been fantastic and we owe them for our last two matches. For different reasons we haven't come out on the right side of a win, now we owe them that.
"Any team that are coming to Dens Park, I don't just want them to play the team on the pitch, I want them to play the crowd in the stands as well.
"I want to make Dens Park that place, we did that last year and there's an opportunity on Saturday to make sure we get back to that and reward the home support with the win.
"I'm expecting a tough game but a really good atmosphere, because St Johnstone always come up here with good numbers."
