You can watch all the goals again below and there will be highlights and analysis on Sportscene on BBC One Scotland at 23:20 BST.
Video content
Video caption: Watch highlights as Celtic beat Inverness CT to win the Scottish CupWatch highlights as Celtic beat Inverness CT to win the Scottish Cup
'There will be a time for questions on my future'
FT: Celtic 3-1 Inverness CT
More typically robust responses from Ange Postecoglou about his future...
Asked on BBC Sportsound why he can't stamp out speculation around a possible departure, he says: "Because I think I deserve to enjoy this moment just like everyone else. I worked really hard for this moment.
"I owe it to my friends and to my family. I get why you have to ask the question, but from my perspective, I deserve to enjoy this moment.
"There will be a time to come for those kinds of questions but for now, we've grasped a small chunk of history for ourselves and I'm not going to pass that by by being side tracked."
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Dodds hopes to stay on at ICT
FT: Celtic 3-1 Inverness CT
Losing manager Billy Dodds is soon out of contract at Inverness Caledonian Thistle.
He says he hopes to stay on but explains the situation is "ongoing" and he won't elaborate.
He's proud of his players, saying it was "a hell of a performance", adding that it's been a "positive, if not successful" season.
'Postecoglou a breath of fresh air'
FT: Celtic 3-1 Inverness CT
Leanne Crichton
Ex-Scotland international on BBC Radio Scotland
It's been a real breath of fresh air since Postecoglou came to Scottish football.
Everything about him - his leadership qualities, the brand of football he plays. He's shown it's not impossible to come as a one man band and have success, which is the most impressive thing for me.
'I will enjoy it for as long as I can'
FT: Celtic 3-1 Inverness CT
Of course, every Ange Postecoglou interview ends with a question about his future...
"I'm going to be a little bit selfish," he says. "People around me have all worked hard to enjoy this moment, so that's what I'm going to do. I will enjoy it for as long as I can before people drag me away to talk about other things."
'We want to be relentless in every way'
FT: Celtic 3-1 Inverness CT
Celtic manager Ange Postecoglou tells BBC Scotland: "I'm very proud. It's been an extraordinary season by the boys. We caught people by surprise last year. It's always hard defending things. The challenge was, 'can we be better this year and we push on?'
"We've won the trophy that eluded us last year, so it feels great, it's the last day of the season. It's an occasion and it was nice to finish it off that way.
"When you think about it, it is pretty relentless. When you play for one of the big clubs, you want that expectation. That's why you come, you want to embrace it. When it's gone, you crave it. You want to be in the position to win things.
"That does come with a great responsibility but I've loved every minute of the way the group has embraced it. It's easy for me as the manager to say that, but the players are the ones who have to perform every week under that pressure.
"We want to be relentless in every way. That's our mantra from day one. I see it every day in training. It's a great group led brilliantly by Callum and the other senior boys. They go down as history-makers now.
"You always get an underdog story. There's always a bit of anxiety and we were a bit nervous at times. We were trying to overdo things, not keeping it simple. Credit to the lads, they always find a way of getting it done."
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Post update
FT: Celtic 3-1 Inverness CT
And now let's hear from the winning manager as he takes time to join the BBC Scotland team at Hampden.
Celtic goalkeeper Joe Hart tells BBC Scotland: "It's really special. It's a great bunch of players, great support staff, a fantastic family football club and we love them to pieces.
[On the emotional scenes at full-time] "There's a pretty clear message, isn't there?
"Look, when you're around amazing footballers and top managers, the noise is always there. That's all it is, noise. Today was the focus, all week.
"Whatever happens now, will happen. It's the world of football. It's an interesting place, full of speculation and rumour."
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Three trophies in six months for Johnston
FT: Celtic 3-1 Inverness CT
Celtic defender Alistair Johnston tells BBC Scotland: "I've been lucky enough to have been here for six months and lift three trophies. It's unbelievable to have won the treble in my first half-season here.
"To do it for this fanbase, you can hear how loud they are. It's a special group of players and a special fanbase. We're just happy we can perform for them.
"In cup football, it's all about getting the result and that's what we did."
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Knock-on bonus for Aberdeen & Hibs
FT: Celtic 3-1 Inverness CT
Celtic's triumph is good news for Aberdeen and Hibernian.
The Europa League play-off spot for the cup winners goes to the Dons since Premiership winners Celtic and runners-up Rangers are bound for the Champions League.
With Aberdeen nudged up, Hibs, who finished in fifth, now get a crack at the Europa Conference League qualifying rounds.
'History makers again'
FT: Celtic 3-1 Inverness CT
Celtic full-back Greg Taylor tells BBC Scotland: "More history for this club. It's unbelievable.
"The badge and the club demand success. We're honoured to be in the position to give it to the fans. It's a brilliant, brilliant day. History makers again.
"The success we've had has been outstanding. We really shouldn't be take it for granted because it's difficult to win one trophy, let alone three. It's a real honour.
"We'll build for next season. He [Postecoglou] has been so good for us. We love working with him, so hopefully that can continue."
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Jota enjoys 'special feeling'
FT: Celtic 3-1 Inverness CT
Celtic goalscorer Jota tells BBC Scotland: "It's something that we started the year aiming for. On the last day, we've done it. I'm so proud of everyone, there was a lot of hard work behind the scenes.
"It's always a special feeling to score in the final. I'm very grateful for this, and I thank everyone who helped me.
"As football players, we want to give happiness to the fans and the big family that Celtic is, and we want to keep doing it."
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Still singing...
FT: Celtic 3-1 Inverness CT
Nick McPheat
BBC Scotland at Hampden
The Celtic fans are still singing about Ange Postecoglou.
They absolutely adore their manager. No wonder. The Australian came in as an unknown to many.
Two years later, he has cleaned up every domestic honour in the country.
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'We've got our trophies back'
FT: Celtic 3-1 Inverness CT
Celtic captain Callum McGregor tells BBC Scotland: "It's hard [to sum up my emotions]. It was a really tough game, finals are never easy - there are a lot of emotions, anxiety and everything.
"First half we weren't great, but we got the goal which was excellent. In the second half we go up a little. We put oursleves under pressure, but another comeback to score again.
"This place is rocking, the fans were outstanding. The success is for these guys.
"Defeats and disappointment are the one thing that is guaranteed. You have to bounce back and pick yourself up. The manager came in and the boys that were left felt like we had something to prove.
"It's taken us two years to get our trophies back. We need to enjoy it, the boys have worked hard."
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Callum McGregor celebrates the second Celtic goalImage caption: Callum McGregor celebrates the second Celtic goal
'Postecoglou exceeds expectations'
FT: Celtic 3-1 Inverness CT
Pat Bonner
Former Celtic goalkeeper on BBC Sportsound
Ange will celebrate with his players and his staff.
When he came to this club, he wouldn't have expected to have that much success. Some of the players have been tremendous under him.
Today wasn't a vintage display, but they got the job done.
Rousing sing-song
FT: Celtic 3-1 Inverness CT
Nick McPheat
BBC Scotland at Hampden
Quite the rendition of You'll Never Walk Alone in the glorious Glasgow sunshine, which has started to come across the lid of Hampden.
Ange Postecoglou and his players stood back and absorbed every second of it. The Celtic boss actually looks quite emotional.
Scottish Cup number 41 for Celtic
FT: Celtic 3-1 Inverness CT
This means 11 of the 24 Scottish Cups played in the 21st Century have been won by Celtic.
It extends their record tally of triumphs in this competition to 41.
'You have to earn these trophies'
FT: Celtic 3-1 Inverness CT
Neil McCann
Former Rangers winger on BBC Scotland
The standards of this club is so high. A lot of people down south might think this is easy done. 'Ange has a huge budget, they're Celtic they should win every trophy.' It's not as easy as that. You aren't given them, you have to earn them all the time.
Inverness pushed them, scored a great goal. But ultimately, the quality that Ange can delve into, the superior fitness as well... Celtic never stopped the rhythm. Today, they were a cut above.
Live Reporting
Colin Moffat
All times stated are UK
Video caption: Watch highlights as Celtic beat Inverness CT to win the Scottish CupWatch highlights as Celtic beat Inverness CT to win the Scottish Cup 
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SNSCopyright: SNS Callum McGregor celebrates the second Celtic goalImage caption: Callum McGregor celebrates the second Celtic goal 
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Latest PostWATCH: Celtic seal treble with Scottish Cup win
FT: Celtic 3-1 Inverness CT
You can watch all the goals again below and there will be highlights and analysis on Sportscene on BBC One Scotland at 23:20 BST.
Video content
'There will be a time for questions on my future'
FT: Celtic 3-1 Inverness CT
More typically robust responses from Ange Postecoglou about his future...
Asked on BBC Sportsound why he can't stamp out speculation around a possible departure, he says: "Because I think I deserve to enjoy this moment just like everyone else. I worked really hard for this moment.
"I owe it to my friends and to my family. I get why you have to ask the question, but from my perspective, I deserve to enjoy this moment.
"There will be a time to come for those kinds of questions but for now, we've grasped a small chunk of history for ourselves and I'm not going to pass that by by being side tracked."
Dodds hopes to stay on at ICT
FT: Celtic 3-1 Inverness CT
Losing manager Billy Dodds is soon out of contract at Inverness Caledonian Thistle.
He says he hopes to stay on but explains the situation is "ongoing" and he won't elaborate.
He's proud of his players, saying it was "a hell of a performance", adding that it's been a "positive, if not successful" season.
'Postecoglou a breath of fresh air'
FT: Celtic 3-1 Inverness CT
Leanne Crichton
Ex-Scotland international on BBC Radio Scotland
It's been a real breath of fresh air since Postecoglou came to Scottish football.
Everything about him - his leadership qualities, the brand of football he plays. He's shown it's not impossible to come as a one man band and have success, which is the most impressive thing for me.
'I will enjoy it for as long as I can'
FT: Celtic 3-1 Inverness CT
Of course, every Ange Postecoglou interview ends with a question about his future...
"I'm going to be a little bit selfish," he says. "People around me have all worked hard to enjoy this moment, so that's what I'm going to do. I will enjoy it for as long as I can before people drag me away to talk about other things."
'We want to be relentless in every way'
FT: Celtic 3-1 Inverness CT
Celtic manager Ange Postecoglou tells BBC Scotland: "I'm very proud. It's been an extraordinary season by the boys. We caught people by surprise last year. It's always hard defending things. The challenge was, 'can we be better this year and we push on?'
"We've won the trophy that eluded us last year, so it feels great, it's the last day of the season. It's an occasion and it was nice to finish it off that way.
"When you think about it, it is pretty relentless. When you play for one of the big clubs, you want that expectation. That's why you come, you want to embrace it. When it's gone, you crave it. You want to be in the position to win things.
"That does come with a great responsibility but I've loved every minute of the way the group has embraced it. It's easy for me as the manager to say that, but the players are the ones who have to perform every week under that pressure.
"We want to be relentless in every way. That's our mantra from day one. I see it every day in training. It's a great group led brilliantly by Callum and the other senior boys. They go down as history-makers now.
"You always get an underdog story. There's always a bit of anxiety and we were a bit nervous at times. We were trying to overdo things, not keeping it simple. Credit to the lads, they always find a way of getting it done."
Post update
FT: Celtic 3-1 Inverness CT
And now let's hear from the winning manager as he takes time to join the BBC Scotland team at Hampden.
'Hope it's not the end for Postecoglou'
FT: Celtic 3-1 Inverness CT
'A pretty clear message' - Hart
FT: Celtic 3-1 Inverness CT
Celtic goalkeeper Joe Hart tells BBC Scotland: "It's really special. It's a great bunch of players, great support staff, a fantastic family football club and we love them to pieces.
[On the emotional scenes at full-time] "There's a pretty clear message, isn't there?
"Look, when you're around amazing footballers and top managers, the noise is always there. That's all it is, noise. Today was the focus, all week.
"Whatever happens now, will happen. It's the world of football. It's an interesting place, full of speculation and rumour."
Three trophies in six months for Johnston
FT: Celtic 3-1 Inverness CT
Celtic defender Alistair Johnston tells BBC Scotland: "I've been lucky enough to have been here for six months and lift three trophies. It's unbelievable to have won the treble in my first half-season here.
"To do it for this fanbase, you can hear how loud they are. It's a special group of players and a special fanbase. We're just happy we can perform for them.
"In cup football, it's all about getting the result and that's what we did."
Knock-on bonus for Aberdeen & Hibs
FT: Celtic 3-1 Inverness CT
Celtic's triumph is good news for Aberdeen and Hibernian.
The Europa League play-off spot for the cup winners goes to the Dons since Premiership winners Celtic and runners-up Rangers are bound for the Champions League.
With Aberdeen nudged up, Hibs, who finished in fifth, now get a crack at the Europa Conference League qualifying rounds.
'History makers again'
FT: Celtic 3-1 Inverness CT
Celtic full-back Greg Taylor tells BBC Scotland: "More history for this club. It's unbelievable.
"The badge and the club demand success. We're honoured to be in the position to give it to the fans. It's a brilliant, brilliant day. History makers again.
"The success we've had has been outstanding. We really shouldn't be take it for granted because it's difficult to win one trophy, let alone three. It's a real honour.
"We'll build for next season. He [Postecoglou] has been so good for us. We love working with him, so hopefully that can continue."
Jota enjoys 'special feeling'
FT: Celtic 3-1 Inverness CT
Celtic goalscorer Jota tells BBC Scotland: "It's something that we started the year aiming for. On the last day, we've done it. I'm so proud of everyone, there was a lot of hard work behind the scenes.
"It's always a special feeling to score in the final. I'm very grateful for this, and I thank everyone who helped me.
"As football players, we want to give happiness to the fans and the big family that Celtic is, and we want to keep doing it."
Still singing...
FT: Celtic 3-1 Inverness CT
Nick McPheat
BBC Scotland at Hampden
The Celtic fans are still singing about Ange Postecoglou.
They absolutely adore their manager. No wonder. The Australian came in as an unknown to many.
Two years later, he has cleaned up every domestic honour in the country.
'We've got our trophies back'
FT: Celtic 3-1 Inverness CT
Celtic captain Callum McGregor tells BBC Scotland: "It's hard [to sum up my emotions]. It was a really tough game, finals are never easy - there are a lot of emotions, anxiety and everything.
"First half we weren't great, but we got the goal which was excellent. In the second half we go up a little. We put oursleves under pressure, but another comeback to score again.
"This place is rocking, the fans were outstanding. The success is for these guys.
"Defeats and disappointment are the one thing that is guaranteed. You have to bounce back and pick yourself up. The manager came in and the boys that were left felt like we had something to prove.
"It's taken us two years to get our trophies back. We need to enjoy it, the boys have worked hard."
'Postecoglou exceeds expectations'
FT: Celtic 3-1 Inverness CT
Pat Bonner
Former Celtic goalkeeper on BBC Sportsound
Ange will celebrate with his players and his staff.
When he came to this club, he wouldn't have expected to have that much success. Some of the players have been tremendous under him.
Today wasn't a vintage display, but they got the job done.
Rousing sing-song
FT: Celtic 3-1 Inverness CT
Nick McPheat
BBC Scotland at Hampden
Quite the rendition of You'll Never Walk Alone in the glorious Glasgow sunshine, which has started to come across the lid of Hampden.
Ange Postecoglou and his players stood back and absorbed every second of it. The Celtic boss actually looks quite emotional.
Scottish Cup number 41 for Celtic
FT: Celtic 3-1 Inverness CT
This means 11 of the 24 Scottish Cups played in the 21st Century have been won by Celtic.
It extends their record tally of triumphs in this competition to 41.
'You have to earn these trophies'
FT: Celtic 3-1 Inverness CT
Neil McCann
Former Rangers winger on BBC Scotland
The standards of this club is so high. A lot of people down south might think this is easy done. 'Ange has a huge budget, they're Celtic they should win every trophy.' It's not as easy as that. You aren't given them, you have to earn them all the time.
Inverness pushed them, scored a great goal. But ultimately, the quality that Ange can delve into, the superior fitness as well... Celtic never stopped the rhythm. Today, they were a cut above.
A message for the manager
FT: Celtic 3-1 Inverness CT