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RCB moved up to seventh place with eight points from 11 games and kept their slim playoffs hopes alive. GT, on the other hand, slipped to ninth spot.
Earlier, RCB dished out a disciplined bowling effort to bowl out GT for a modest 147..
Opener Kohli (70 not out) and one-down Jacks (100 not out) shared 166 runs for the unbroken second wicket to overhaul the target of 201 with 24 balls to spare.
Kohli smashed an unbeaten 70 off 44 balls and used the sweep shot to good effect against the Gujarat Titans spinners.
SRH are in third spot with 10 points from eight games, while RCB remain at the bottom with four points from nine matches.
The loss was SRH's third in eight matches and their first at home this season.
Kohli was hauled up for expressing dissent against the umpire's decision.
The ball's projected height was found to be lower than Kohli's waist, leading to the third umpire ruling him out.
RCB were bowled out for 221 in their pursuit of 223 despite three sixes by Karn Sharma off Mitchell Starc in the final over.
Kohli was caught and bowled by Harshit Rana off a full toss for 18 in RCB's chase of 223. Kohli, who was batting outside his crease, was quick to refer for a no-ball.