Chennai, already out of the playoff race, overcame another top-order batting collapse to reach 8-183 for only their third win in 12 matches, and were largely grateful to South African Dewald Brevis's thumping 52 off 25 balls.
The defeat leaves Kolkata sixth, with 11 points and just two games in hand, a perilous position as they have to finish in the top four.
Wrist-spinner Noor Ahmad took the player of the match award for Chennai for having earlier picked up 4-31 on a favourable wicket for spinners to restrict Kolkata at 6-179.
Brevis didn't hold back despite Chennai losing half the side inside the first six overs of the batting powerplay.
Kolkata had Chennai on the mat at 5-60 inside the first six overs despite young IPL debutant Urvil Patel making a rapid 31 off 11 balls that featured four sixes.
With 87 needed off 10 overs, Brevis turned the game, smashing fast bowler Vaibhav Arora for 6-4-4-6-6-4 in one over as he raised his maiden IPL half-century off 22 balls.
Kolakta fought back when Brevis holed out at long-on in the 13th over, but impact player Shivam Dube (45) and MS Dhoni put on 43.
Arora picked up the wickets of Dube and Ahmad in the penultimate over but master finisher Dhoni (17no) ensured Chennai didn't slip up in the last over as he lofted Andre Russell for six over mid-wicket. No.10 Anshul Kamboj then sealed the game with a boundary.
Earlier, Ahmad combined with Ravindra Jadeja (1-34) and Ravichandran Ashwin (0-19) to restrict Kolkata. The three spinners bowled 11 overs between them to finish with 5-84.
Kolkata lost Rahmanullah Gurbaz (11) in Kamboj's first over but Sunil Narine (26) and captain Ajinkya Rahane (48) hit a combined eight fours and two sixes to take it to 1-67 in the six-over powerplay.
But Ahmad had two wickets in his first over when he had Narine stumped and then Angkrish Raghuvanshi edged the left-arm spinner to give Dhoni his 200th catch in the IPL.
Russell provided impetus by smashing three sixes and four boundaries in his 21-ball knock of 38 before Ahmad had the key man caught at long-off in the death overs.
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