Pandya dropped a hard chance to allow Christian to take two runs off the penultimate ball, but even a fumble in the outfield did not give the Australian and Sundar enough time to get back for a third off the last delivery.
The victory was sealed with the final ball of the game, bowled by former Australia global Mitchell Johnson, when Pune could only get two runs when they required four for the win.
In a match that went right down to the last ball, Mumbai registered a one-run victory in the packed Rajiv Gandhi International stadium.
Earlier, braces by bowlers Jaydev Unadkat, Adam Zampa and Christian enabled Pune to restrict Mumbai to 129 for eight. Smith, however, created the platform by giving the ball to Daniel Christian who conceded 27 runs in the 18th and 20th overs despite he has the overs left from Shardul Thakur.
Rising Pune Supergiant chief coach Stephen Fleming has described their outing in IPL 10 as a grand finish to a good competition.
In reply, Ajinkya Rahane struck 44 off 38 balls before he was removed by Johnson to leave Pune on 71-2 after 12 overs. With RPS needing to score 4 runs, MI held their nerves as Washington Sundar was run out for a golden duck.
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"I'm calm now, it was a great game of cricket", Mumbai Indians captain Rohit Sharma said.
Mumbai struggled with pace and spin, scoring just 16 total runs in the first five overs.
Johnson bowled one brilliant over as he picked up wickets of Manoj Tiwary (7) and RPS skipper Steve Smith (51) to script a memorable victory.
Mumbai previously won the IPL in 2013 and 2015. In this chat for iplt20.com, he briefly reflects on the season and 'keeping to Mitchell Johnson in the crucial last over in the grand finale.
Asked about Mumbai Indians' strategy going into the last over with a set Pune captain Steven Smith at the crease, Rohit said: "There was no specific strategy".