Jack Butland was Rangers' Player of the Year but the goalkeeper made a rare mistake to hand Celtic the Scottish Cup trophy.
Hampden Park was rocking ahead of kick-off with both sets of fans chanting loudly.
But the Old Firm derby was far from a classic on the pitch with neither side creating much in front of goal.
Rangers thought they had taken the lead midway through the second half when Abdallah Sima put the ball in the back of the net.
Adam Idah pounced on Jack Butland's mistake
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But referee Nick Walsh disallowed the goal after a VAR check spotted Nicolas Raskins had shoved Joe Hart.
Tensions were rising on the pitch as the prospect of extra time drew nearer with multiple yellow cards being branded as tempers flared.
However, extra time soon looked unlikely after a costly error from Butland in goal.
Paulo Bernardo tried his luck from range with a weak effort that appeared to wrongfoot the Rangers goalkeeper.
Butland ended up parrying the shot back into the middle of the box and straight into the path of Adam Idah.
Rangers had just eight minutes of stoppage time to try and find a leveller with Celtic happy to sit back.
But Philippe Clement's lacked the cutting edge in the final third to make anything happen with Hart largely unchallenged throughout.
The former Manchester City and England goalkeeper was in tears at the final whistle as the Scottish Cup final marked his final game before retirement.
Rangers thought they had taken the lead earlier in the match
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Butland had his hands on his knees at the end as Celtic began their celebrations on the pitch.]
The win also gives Brendan Rodgers the Scottish Cup and Premiership double after holding off Rangers in the league.
After the match, Celtic's match winner Idah was asked whether he wanted to return to the club on a permanent basis.
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Celtic completed the Scottish Cup and league double
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"I said I'd come here to try and score as many goals as I could, I didn't think I'd score a goal like this in front of all these fans. It's an amazing feeling," he said.
"It's hard to put into words. Anyone here can see what it means to the fans, to the players. I've always supported Celtic, it's amazing.
"Football is tricky. I love it here, I've said it previously, it's a great club but I'm still contracted to Norwich. I'll go back in the summer and who knows?"