Chasing 130 to win, Pune looked primed for their maiden IPL title with skipper Steve Smith leading the charge and the team needing 11 runs off the final over with seven wickets in hand.
Australia's former Ashes hero Johnson dismissed Manoj Tiwary and Smith off successive deliveries while Dan Christian ran himself out in the final ball of the over which yielded nine runs.
Smith topscored for his side with 51 off 50 balls while Ajinkya Rahane made 44 upfront but Pune never got the late assault despite having wickets in hand and finished 128-6.
Johnson claimed 3-26 while his pace colleagues Jasprit Bumrah (2-26) and Lasith Malinga, who went wicketless but conceded only 21 runs in his four overs, were equally impressive in defending a modest total.
Mumbai went into the match having lost their previous three matches, including Tuesday's qualifier, against Pune this season and the 129-8 they managed on Sunday looked far from an intimidating score.
Mumbai skipper Rohit Sharma opted to bat but they looked in trouble after Jaydev Unadkat (2-19) dismissed both the openers cheaply to deny them a strong start.
Mumbai crawled to 56-3 at the halfway stage of their innings when Adam Zampa struck twice in the same over to wreck their hopes of a big total.
The Australian leg-spinner dismissed Rohit with the first delivery of his over with Shardul Thakur taking a well-judged catch near the rope and the spinner's sixth sent back Kieron Pollard.
Mumbai were reeling at 79-7 but Krunal Pandya topscored for them with a belligerent 47 before falling to the final delivery of the innings.
"It was a great game of cricket and I hope the crowd enjoyed it," Rohit said.
"To defend such a modest total, the first thing you need is self-belief and I told the boys we can do it. It was a great effort by the team, hats off to them."
(Reporting by Amlan Chakraborty in New Delhi; editing by Pritha Sarkar)
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