Chasing 116, Shubman Gill made 57 and Nitish Rana a crucial 25 before Dinesh Karthik hit a 12-ball 18 to win the game with two balls left.
SRH made 115 for eight with Kane Williamson top scoring with 26. Kolkata remain fourth, but on 12 points as they edge clear of the chasing pack.
Kolkata’s win was built on the partnership of 55 from 59 between Gill and Rana which, while it may not win any awards for entertainment, was match winning.
Gill, who opened the batting for India against Australia and England last winter, showed his class as he latched on to the bad balls – scoring 23 runs via the pull shot – and kept out the good ones.
Gill fell with 23 needed and Rana 10, but Karthik used all his experience, and skill, to hit three crucial boundaries and secure the win.
Sunrisers Hyderabad: 115/8 in 20 overs (Williamson 26, Southee 2/26, Mavi 2/26)
Kolkata Knight Riders: 119/4 in 19.4 overs (Gill 57, Rana 25, Holder 2/32). (BBC)