In the Super Over, Punjab made five for the loss of one wicket and scores were tied as Mumbai too made five for no loss.
However, Mohammed Shami bowled an equally incredible over as KXIP defended the five runs to take the match to a second Super Over.
Bumrah (3/24) snared three crucial wickets during the regulation 20 overs after opener Quinton de Kock (53 off 43) hit his fourth half-century of the season to restrict KXIP.
Pollard then clobbered Arshdeep for successive sixes and Coulter-Nile hit two fours as MI amassed 22 runs in the 18th over.
Batting in the unusual position of four in the order, Warner made an unbeaten 47 off 33 balls as Sunrisers managed to get 17 runs off the final over to level the scores. Agarwal finished the job with back-to-back boundaries to bring an historic day of IPL action to an end after three Super Overs across the two games.
MUMBAI INDIANS: Rohit Sharma b Arshdeep 9 (8b, 2x4), Quinton de Kock c Mayank b Jordan 53 (43b, 3x4, 3x6), Suryakumar Yadav c M. Ashwin b Shami 0 (4b), Ishan Kishan c M. Ashwin b Arshdeep 7 (7b, 1x4), Krunal Pandya c Hooda b Bishnoi 34 (30b, 4x4, 1x6), Hardik Pandya c Pooran b Shami 8 (4b, 1x6), Kieron Pollard (not out) 34 (12b, 1x4, 4x6), Nathan Coulter-Nile (not out) 24 (12b, 4x4); Extras (b-4, lb-2, w-1): 7; Total (for six wkts.in 20 overs): 176.
Shami replicated what his Indian teammate had done earlier and kept things tight, giving just three from first four balls.
Kings XI lost the in-form Agarwal early in the reply, bowled by Bumrah, before Rahul shared useful partnerships with Gayle (24 from 21) and Nicholas Pooran (24 from 12) to keep the IPL strugglers in contention.
When part-time spinner Deepak Hooda was given an over, his short delivery was sent out of the park by de Kock, while M Ashwin conceded 16 runs in his first two overs. But KL Rahul's sublime 55-ball 77 kept KXIP in the game. It soon swung back, though, as Hooda and Jordan took 13 from the penultimate over from Coulter-Nile to leave themselves needing nine from the last.
Rahul started the proceedings with a single before Pooran got out on the second ball.
The laws dictate that a second Super Over requires a change of bowler and batsmen so Jordan took the ball for Kings XI with Pollard, perhaps surprisingly overlooked for the first one-over shootout, and Hardik Pandya in for Mumbai. Chris Jordan bowled the second Super Over for KXIP. The Universe Boss hammered a six over the long-on on the first ball and then ran a single.
The fifth ball was a dot before Mayank Agarwal put in a stunning effort to save a six. Instead the throw came in, low and hard for the wicketkeeper to deal with but Rahul dived and flicked the ball onto the stumps to run out the diving De Kock by half a foot.