Asked to bat, RCB were helped by de Villiers' knock that contained two sixes and five fours. Captain Virat Kohli was the next best scorer with a 37-ball 41.
But pace spearhead Boult pegged back RCB by taking the wickets of de Villiers, Sean Abbott and Harshal Patel in the innings' penultimate over.
Bhuvneshwar Kumar then grabbed two wickets in the final over as RCB's innings ended with a ball to spare.
Gayle started the onslaught by lofting Boult straight over his head for a massive six in only the third ball of the innings.
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Kohli then majestically drove Kumar past mid-off, bringing the packed home crowd on its feet with his strokeplay.
Gayle followed it up with three more boundaries before medium pacer Praveen Kumar had him caught at deep square leg while going for a wild flick. The Jamaican hit three fours and a six during his 16-ball 21.
Captain Kohli kept the spectators excited with two sixes.
However, it was SunRisers Hyderabad who had more to cheer about as Ravi Bopara, with his military medium pace bowling, sent back the dangerous Kohli (41) and Mandeep Singh off successive deliveries in the 12th over. Hyderabad skipper David Warner pulled off a stunning one-handed catch at cover to send back Mandeep, leaping to right with both his feet off the ground.
But RCB still had De Villiers to bank on and the big-hitting South African tried to deliver before falling to Boult.
Before that, de Villiers smashed Bopara for three successive boundaries.