The teams were tied on 162 runs after 20 overs thanks to a last-ball six by Hyderabad's Manish Pandey and Mumbai skipper Rohit Sharma said Bumrah's cool head made all the difference in the end as the three-times champions prevailed.
"He held his nerve in the super over where it can go any way," Rohit said in the post-match presentation ceremony.
"I talk about him every now and then. He's been brilliant. He keeps improving in every game that he plays and he is hungry for success, that's for sure.
"He wants to go out in every game and express himself... he understands he's the lead bowler for us."
Bumrah's fine form bodes well for the Indian team ahead of their World Cup campaign in England and Wales where conditions are expected to suit fast bowlers more.
The 25-year-old, who has 15 wickets for Mumbai this season, said positive thinking and clarity of mind were the secrets of his success while bowling at the death.
"Sometimes the mind can get too negative, so I focus on the ball I have to bowl and try to keep the clarity," Bumrah said.
"There's always pressure but you don't try to think about it. Whenever I feel pressure, I think of what the team needs of me. I just focus on one ball at a time.
"Whenever you're bowling in tough situations you try to back yourself as much as you can. I try to keep things simple so that has always worked for me."
Second-placed Mumbai joined champions Chennai Super Kings and Delhi Capitals in qualifying for the knockouts. Four teams are in contention for the final spot with fourth-placed Hyderabad the best-placed to progress.
(Reporting by Shrivathsa Sridhar in Bengaluru; editing by Sudipto Ganguly)
