Sunrisers seemed out of reach after Moises Henriques (57 off 22 balls) and fellow Australian David Warner (52no off 32 balls) shared a stand of 103 in 7.1 overs to propel the home side to 135-3 in their permitted 11 overs.
But after a further shower, Bangalore careered to their revised target of 81 runs off six overs with one ball remaining after Gayle pillaged 35 off 10 balls and skipper Kohli helped himself to 44no off 19 balls.
Kohli finished the match in style with a straight six although he was indebted to opposing captain Warner who accidentally stepped on the rope while completing what would otherwise have been catch. The result propelled Bangalore up to second place in the table, within touching distance of a place in the play-offs.
After a delayed start, it was Henriques and Warner to the fore. The pair came together after Shikhar Dhawan holed out to mid-on for eight off the bowling of David Wiese, but that was as good as it got for the Bangalore attack.
Warner, for once, was rather overshadowed as Henriques raced to his half-century off 20 balls ??? although he enjoyed three large slices of luck on the way.
The 28-year-old had 26 to his name when he was put down by Mandeep Singh at long-off ??? the ball slipping away for four in a seventh over that cost 21 runs in total.
Harshal Patel then missed a simple run out chance in his follow-through ??? his under-arm shy straying wide of the stumps with the batsman well short of his ground ??? and two balls later Sarfaraz Khan shelled a straightforward chance at point off Mitchell Starc when Henriques had 47.
The Australian???s charmed life came to an end when he went after a wider Wiese delivery and edged behind.
There was just time for a brief cameo from Englishman Eoin Morgan, who struck one six before top-edging the final ball of the innings from Starc to long-off.
Gayle struck Dale Steyn???s first two balls for 10 runs ??? the South African???s first over going for 17 in total as Hyderabad???s chase got off to a flying start.
Bhuvneshwar Kumar???s first over suffered greater punishment as Gayle plundered three fours and a brace of sixes to propel the score to 41 off 12 balls.
Something had to give and it was Gayle, who mowed Henriques to Dhawan in the deep to depart for 35 off 10 balls.
Henriques then found himself on a hat-trick as AB de Villiers struck his first ball down the ground but without sufficient power to clear Steyn.
Kohil cut for a single to deny Henriques a third wicket in as many balls and then had a let-off on 11 as Karn Sharma could not hold a tough chance running back off his own bowling.
With 23 needed off 12 balls, Mandeep flicked Praveen Kumar to deep square leg where Morgan took a simple catch to leave Bangalore hopes resting with Kohli.
Bhuvneshwar Kumar took the ball with the batting side needing 13 off six ??? Kohli settling for a single off the first ball before Dinesh Karthik was run out for two.
The captain???s self-belief was justified as he pocketed back-to-back boundaries before sealing the match in the grand manner, all be it with some assistance from Warner.