Mumbai were initially thankful to back-end cameos from Tirumalasetti Suman (36) and Kieron Pollard (28) for stretching their total to 160, after starting the 16th over at just 97.
They fared much better with the ball - Lasith Malinga took three for 25 - as Pune stuttered to 84 for four and eventually ran out of overs to play despite Manish Pandey's 59 off 47 balls and Robin Uthappa's 34.
Sent in to bat, Aiden Blizzard could not make the most of his recall, out early for just six, but captain Sachin Tendulkar and Ambati Rayudu (27) helped Mumbai keep a steady grip.
It needed a diving catch from Jesse Ryder to dismiss Tendulkar for 24 after a 40-run stand, and skipper Yuvraj Singh had a second in his next over as Rayudu holed out to long-off.
Rohit Sharma was next to depart, followed a few overs later by Suman, whose gutsy 36 included three boundaries and as many sixes.
Pollard then provided the late fireworks with his 21-ball salvo - he was dismissed on the final ball - as Mumbai picked up 46 off the last three overs.
Pune's batting was jittery at the start as Ryder departed first ball to Harbhajan Singh and Graeme Smith was ousted in the fourth over by Munaf Patel's bouncer.
Pandey attempted to keep Pune in the hunt but found little support at the other end, as Abhishek Jhunjhunwala (10) and Yuvraj (20) fell to Pollard and Malinga respectively.
Pandey eventually got to his half-century in 43 deliveries courtesy of a few big hits late on, but Malinga's 19th-over double strike removed him and Mithun Manhas as Pune fell short having reached 139.