Mumbai could not find the 10 runs they needed for victory in the final over despite having only lost three wickets prior to it. Kumar took two additional wickets then sent down a perfect yorker on the final ball to secure the win with personal figures of 3-27.
Former Australia captain Ponting won the toss elected to field, and his Mumbai side's attack - led by compatriot Mitchell Johnson - ensured a slow run-rate early on for Bangalore.
However they could not get Chris Gayle out. He hit 92 off 58 balls including 11 fours and five sixes, while his team-mates floundered around him.
The West Indies legend saw his side reach 156 for 5, with Virat Kohli the next-best batsman with just 24.
Mumbai's initial reply rumbled along nicely, with openers Sachin Tendulkar (23) and Ponting (28) reaching 50 in seven overs before both departed.
The run-rate was not to improve in the middle overs, however, with Dinesh Karthik (60) finally accelerating the innings in the 17th over with a boundary and three consecutive sixes.
Karthik and Ambati Rayudu (18) fell in the final over, with Kieron Pollard (5no) getting on strike with eight runs needed off two balls - the first he sent for a four, with the ball agonisingly dropping inches short of the rope, but the second yorker he could only send to long-on.
It was the second match of the IPL, with defending champions Kolkata Knight Riders winning the first match against Delhi Daredevils by six wickets on Wednesday.
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