SRH skipper Kane Williamson won the toss and elected to field.
Bhuvneshwar Kumar took 3-26, while Shakib Al Hasan (2-21) and Billy Stanlake (2-21) were also impressive with the ball in Kolkata.
"It's nice to get out there and take a few wickets". We thought it was going to come through nicely.
"Obviously nice to win three in a row, and this surface required some adapting, but the real highlight was our fielding tonight", Hyderabad captain Williamson said at the post match presentation.
Williamson and Shakib eased the nerves - Williamson notching his 18th T20 fifty from 43 balls and Shakib passing 4,000 runs in the format - and even though they fell towards the death, Yusuf Pathan (17no of 7) pulled Russell over deep square for the match-winning six with an over to spare. The SRH bowlers were supported admirably by their fielders, typified by two stunning catches from Manish Pandey, a former Knight Rider himself, the second of which helped remove the risky Andre Russell.
Highly-rated leg-spinner Rashid Khan went wicketless, leaking 31 runs.
For KKR, Sunil Narine picked up 2 wickets while Kuldeep Yadav, Mitchell Johnson and Piyush Chawla all getting one wicket each. Chris Lynn looked good for his 34-ball 49 but was taken out by Shakib, and Karthik's 29 towards the end of the innings was the only other significant contribution with the bat.
SRH registered its third straight win in the season to consolidate their position at the top of the points table while KKR suffered its second successive defeat and are now placed fourth.