Warner has returned to the franchise this season but SRH have retained Williamson as the skipper.
KKR needed 53 in the last three overs and they won with two balls to spare thanks to the West Indian Russell.
Vijay Shankar also made a fine return to the Oragne Army with a quick-fire unbeaten 24-ball 40 (2x4, 2x6) after Warner departed in the 16th over. With destructive openers - Sunil Narine and Chris Lynn - followed by a solid Indian middle order in Robin Uthappa, Shubhman Gill and Nitish Rana. Warner, who is coming towards the end of his 12-month global suspension for the ball-tampering scandal in Cape Town, modelled his innings on his trademark aggression hitting nine boundaries and three sixes during his 53-ball knock.
Earlier, the Knight Riders had won the toss and elected to field and Warner had opened the batting with England's Jonny Bairstow.
Russell, a veteran of T20s, took Siddharth Kaul for 19 runs in the 18th over, followed by 21 off Sunrisers stand-in skipper Bhuvneshwar Kumar, to bring the equation down to 13 off the last over. Warner made a stunning 53-ball 85 which included 9 boundaries and 3 sixes.
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The 32-year-old has now scored 37 IPL half-centuries, more than anyone else in the competition's history, despite missing last year's tournament following the fallout from his global ban. The opener started hitting all over the park and literally punished all the bowlers of KKR.
With 37 half centuries, Warner has overtaken former cricketer Gautam Gambhir with whom he had till now jointly held the record of the most number of 36 half-centuries. "We just felt there was no need to make any change from a year ago".
But it hardly seemed to matter as in-form all rounder Vijay Shankar showed his confidence by going down on one knee to slog sweep Narine over midwicket for a superb six.
SRH were dealt with a major blow after Kane Williamson, who was the last season's top run scorer, was ruled out for the initial part of the tournament owing to injury.
Shubman Gill, however, delivered the finishing touch by blasting Shakib Al Hasan for two sixes to cap a stunning comeback victory. "It's not a long term injury, whether he plays, we will take a call tomorrow", Moody said on the eve of the match.