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Having posted a modest 149-8, butter-fingered Mumbai found themselves on the brink of defeat but Lasith Malinga conceded just seven runs off the final over to seal Mumbai's fourth IPL title.
Chennai opener Shane Watson rode his luck to smash the match's only half-century but his 80 could not prevent his team going down to Mumbai for the fourth time in this year's IPL.
Earlier, Quinton de Kock smashed four sixes, including three in a Deepak Chahar (3-26) over, in his 29 but Mumbai lost both their openers, including skipper Rohit Sharma, in successive overs after opting to bat.
De Kock fluffed his pull shot off Shardul Thakur, while Deepak Chahar foxed Rohit with a knuckle-ball with Chennai captain Mahendra Singh Dhoni taking both catches.
Boundaries soon dried up and pressure mounted on Mumbai who kept losing wickets in their stop-start innings.
Leg-spinner Imran Tahir claimed a couple to finish as the tournament's leading wicket-taker while Thakur took a brilliant running catch off his own bowling to dismiss Krunal Pandya.
Krunal's brother Hardik Pandya (16) did hardly any better but Kieron Pollard smashed three sixes in his 41 not out off 25 balls to help them to a modest total.
Faf du Plessis (26) and Watson survived a couple of mix-ups in an opening stand of 33 when Chennai replied.
Watson struggled to find gaps initially but hung on, aided by generous doses of luck.
Rahul Chahar dropped Watson twice, including once off his own bowling, while Malinga could not place himself under a swirling skyer when the all-rounder was on 31.
Mumbai roared back into the contest by claiming three quick wickets, including that of Dhoni who perished trying to steal a second run only to be beaten by a direct throw.
Chennai plundered 20 runs off the 16th over and as many from the 18th when Watson smashed Krunal Pandya for three successive sixes.
There was more drama in the final over, with Watson running himself out after another mix-up, his exit leaving his team needing four off the last two deliveries.
Thakur collected two off the fifth ball but Malinga then trapped him lbw with the final delivery of the match to trigger wild celebrations in the Mumbai camp.