Dropped on eight in the second over by Mitchell McClenaghan off his own bowling, in-form opener Gambhir -- who made 59 off 45 balls -- stitched crucial stands of 69 for the first wicket with Robin Uthappa (36 off 20) and 44 for the third wicket with Suryakumar Yadav (21).
Later, big-hitting West Indian Andre Russell hit a quick 22 off 16 balls with three fours while Yusuf Pathan remained unbeaten on 19 off 8 balls which included four boundaries to get KKR past the 170-run mark.
For MI, playing their final game of the season at this ground, McClenaghan(1 for 33), his New Zealand teammate Southee (2 for 38), seasoned off-spinner Harbhajan Singh (1 for 31) and Hardik Pandya (1 for 15) were the wicket takers.
KKR were off to a breezy start with the openers raising 69 in 46 balls.
The visitors, champions in 2012 and 2014, raced to 48 without loss in five overs and 59 at the end of the first six power-play overs with Gambhir giving the initial thrust before his partner Uthappa followed suit.
Uthappa clouted Jasprit Bumrah over the straight field for a six and pulled Harbhajan over mid-wicket for another maximum.
The off-spinner exacted revenge in his second over by having the Karnataka Ranji player caught at long on by Kieron Pollard after a 20-ball essay in which he hit two sixes and one four.
Hardik then struck quickly to pack off Shakib Al Hasan for 6 to put brakes on KKR's rousing start.
KKR captain Gambhir picked up his attacking reins in the very next over from Krunal Pandya to cart the left-arm spinner over long on for his first six of the innings and followed it with another lofted four towards the same region to reach his third 50 of the season in 39 balls.
He eventually fell to McClenaghan after mistiming another pull shot only to hand a tame catch to Pollard.
Gambhir's departure cut short his third wicket stand with Suryakumar Yadav in the 14th over. KKR the added 44 runs in the last four overs.
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