Put into bat, Bengaluru posted a modest 190-9 before returning to restrict Punjab to 184-7 at the Narendra Modi Stadium.
Virat Kohli top-scored for his side with 43 but it was their lion-hearted bowling which secured their memorable victory.
Kohli was in tears once he realised Bengaluru were about to shed the underachiever's tag that has stuck to them since the inaugural 2008 edition of the league.
Put in to bat at a heaving Narendra Modi Stadium, Bengaluru could not stitch together significant partnerships and fell short of the 200 mark.
Kohli anchored their innings but Bengaluru could never really break loose. They were 87-2 at the midway stage but with enough overs left for other players to bat around Kohli.
Azmatullah Omarzai deceived Kohli with a bouncer and took a running, tumbling return catch to dismiss the batter.
Jitesh Sharma's 24 off 10 balls injected some momentum into Bengaluru's stop-start innings and Liam Livingstone made 25 but the side could not really capitalise in the death overs.
Punjab's Kyle Jamieson claimed 3-48, while his new ball partner Arshdeep Singh bowled an excellent three-wicket final over, conceding only three runs.
Punjab got off to a rollicking start in their chase and opener Prabhsimran Singh, then on nine, was lucky to be spilled in the deep by Romario Shepherd.
Josh Hazlewood, the luckless Bengaluru bowler, did not have to wait long for success though, as Phil Salt took a stunning catch in the deep to remove Priyansh Arya (24) and break the 43-run opening stand.
Krunal Pandya removed Prabhsimran and Shepherd redeemed himself by dismissing Punjab's in-form captain Shreyas Iyer to turn the match on its head.
Left-arm spinner Pandya put Bengaluru on top when he dismissed the dangerous Josh Inglis (39) and Bhuvneshwar Kumar effectively sealed the match in Bengaluru's favour when he dismissed Nehal Wadhera and Marcus Stoinis in the same over.