The Indian pacer's masterful 5-21 won him the player-of-the-match award as the star-studded RCB line-up made 196-8 after Hardik Pandya won the toss and decided to field first.
Bumrah removed fellow India superstar Virat Kohli (three) before rookie pacer Akash Madhwal dismissed England's Will Jacks (eight) in the fourth over to leave RCB reeling at 23-2.
RCB skipper Faf du Plessis rebuilt the innings on a placid Mumbai track with young Rajat Patidar, who hit an impressive 26-ball 50 with four sixes and three fours.
South African fast bowler Gerald Coetzee took Patidar's wicket while Australian superstar Glenn Maxwell (nought) failed again to leave RCB in trouble 108-4 in the 13th over.
Du Plessis fell to Bumrah, who returned to demolish the RCB middle-order, for a 40-ball 61.
Bumrah also removed Mahipal Lomoror (nought) on the very next ball to leave the visitors at 153-6 in the 17th over.
A late blitz by veteran Dinesh Karthik, who hit an unbeaten 23-ball 53, took RCB to 196-8 as Bumrah took two more wickets - rookies Saurav Chauhan and Vijaykumar Vyshak - to finish his match-defining spell.
Speaking after the game, Bumrah said that the T20 format "is very harsh for the bowlers".
"I try to not be a one-trick pony, I have worked on this early in my career. People start to line you up. I want to have different skills."
He added: "I will never say I wanted to take a five-for. The wicket was sticky and I am very happy with my contribution."
Mumbai openers Kishan and Rohit Sharma set up the chase with a century partnership before Kishan, who hit five sixes in his 34-ball 69, fell in the ninth over with the team at 101-1.
Sharma, who scored a steady 24-ball 38, built another partnership with Suryakumar Yadav, who hit a 19-ball 52.
Sharma was removed by Jacks with Mumbai at 139-2. But Yadav's blitz propelled the home team to 176 in the 14th over before rookie Vyshak took his wicket.
Pandya and young Tilak Verma finished the job for the home team in the 16th over.
Pandya said that he was "lucky to have Bumrah" in his side.
"He does this over and over again and every time I ask him he gets the wickets. He has so much experience and confidence," he added.
Pandya also hailed Suryakumar.
"I did tell him, welcome back when he scored the fifty. I have been opposition captain to him as well. Some place he hits, I have never seen any batter hit."
Du Plessis praised Bumrah and said that "he has been the difference in the two innings".
"You see him with the ball in his hand. You want to put him under pressure, but the variety does everyone in."
Mumbai are seventh in the table while RCB are now ninth, just above Delhi Capitals, in the 10-team cash-rich tournament.
