Chasing 146 for their fourth win in a row, Chennai achieved the target with three balls to spare as openers Dwayne Smith (66) and Brendon McCullum (40) laid a solid foundation with their 85-run partnership in 10.5 overs.
Smith and McCullum made batting look easy as they struck the ball effortlessly even as all other batsmen struggled on a difficult wicket.
Chennai were cruising at one stage, needing just 22 runs from the last five overs with eight wickets in hand but young Indian paceman Bhuvaneshwar Kumar changed the complexion of the game by dismissing Smith and Faf du Plessis (0) in consecutive deliveries in the 16th over.
CSK needed six runs off the last over, bowled by Amit Mishra, and skipper Mahendra Singh Dhoni finished it off in style, dismissing a full toss to square leg boundary.
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Chennai had control over the contest for the most part but for the last two overs of the Hyderabad innings when their bowlers took a beating to concede 32 runs and later in the final overs of their own innings.
The Australian batsman held the one end when wickets fell consistently from the other before departing in the fag end.
Finch added 52 runs for the third wicket with Lokesh Rahul (25) and had it not been for some lusty hits by Darren Sammy (23 not out) and Karan Sharma (17 not out) towards the end, Hyderabad would not have posted a decent total.
Sharma hammered paceman Ben Hilfenhaus for two sixes in the penultimate over and Sammy also succeeded in hitting a maximum off Mohit Sharma in the final over.
McCullum entertained the crowd with three shots to the fence and two over the boundary ropes in his 33-ball knock.
Despite losing his partner, Smith went about his business calmly and completed his third half-century of this season with a single off paceman Ishant Sharma. The burly West Indian then clobbered Amit Mishra for a six to raise 100 of the CSK innings.
Suresh Raina hit a sweetly-timed six off Ishant Sharma but the paceman got his revenge by having the left-hander caught in the last ball of that over.