Put into bat on Tuesday, Bengaluru posted a modest 9 for 190 before restricting Ricky Ponting-coached Punjab to 7 for 184 at the Narendra Modi Stadium.
Kohli top-scored for his side with 43 but it was their bowling which secured Bengaluru's memorable victory.
Fast bowler Josh Hazlewood claimed one wicket, but fellow Australians Josh Inglis (39) and Marcus Stoinis (6) both ended up on the losing side.
Bengaluru finished runners-up in 2009, 2011 and 2016, but finally claimed the prize, thanks largely to Krunal Pandya's sensational 2-17 in four overs.
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.
Afterwards, a jubilant Hazlewood said: "The wicket was a little bit up and down. Our guys batted really well to post that score - everyone just chipping in - and the bowlers held their nerve, so great result."
Asked what it would mean to Kohli to end his side's title quest at last, Hazlewood said: "It probably means everything to him. Some of us come and go in the franchises but to be there from the start and to slug it out for 18 seasons and to get a result tonight, I think we'll see a bit of emotion there."
Hazlewood and Inglis were due to fly straight to England after the final for next week's World Test Championship against South Africa. It will have made for an interesting flight.
Put in to bat, Bengaluru fell short of the 200-mark, having been 2 for 87 at the midway stage. Kohli anchored their innings but Bengaluru could never really break loose.
Azmatullah Omarzai deceived Kohli with a bouncer and took a running, tumbling return catch.
Jitesh Sharma's 24 off 10 balls injected some momentum into Bengaluru's stop-start innings and Liam Livingstone made 25 but they never capitalised 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 fast start in their chase and opener Prabhsimran Singh, on nine, was lucky to be spilled in the deep by Romario Shepherd.
Hazlewood, the luckless bowler then, did not have to wait long for a wicket, 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 Inglis and Bhuvneshwar Kumar effectively sealed the match when he dismissed Nehal Wadhera and Stoinis in the same over.
With agencies
7°C