Praveen bowled a sensational spell after inviting hosts to bat as he gave away just eight runs in his four overs and seasoned Mahmood took three wickets at the expense of just 20 runs to restrict hosts to 158 for five.
Bangalore owed their total to Chris Gayle`s blazing 71 off 42 balls since they were struggling at 59 for one in 10 overs at one stage.
Chasing 159, Punjab had some very anxious moments before crossing the line as they lost Azhar Mahmood (2), skipper David Hussey (45) and Abhishek Nayar (2) through freak run outs in a space of eight balls towards the end to almost give away the win to rivals on the platter.
However, Piyush Chawla struck a six in the penultimate ball of the match to ensure that his side jump to fourth place in the table from a dejecting eighth spot.
Nitin Saini (50 off 36 balls), Mandep Singh (43 off 30) and skipper Hussey batted with sheer determination before the unnecessary run outs.
Saini shared a match-winning partnership of 73 runs for the third wicket with Hussey (45 off 29) to lay foundation for the win.
However, Punjab made it tough for themselves as Mahmood lost wicket in the fourth ball of the 17th over and Hussey and Nayar were out in the 19th over bowled by Zaheer Khan, leaving Punjab to score five off six balls.
Vinay Kumar did not concede a run in the first two balls but Paras Dogra managed a couple off the third to make it three runs off three balls.
It became two off two when Dogra took a single in the next ball and then Piyush Chawla ended supense with a big six.
Now Punjab (-0.280) and Mumbai Indians (-0.187) both have 10 points but the latter are placed third due to better Net Run Rate.