Chasing 145 to take an unassailable 2-0 lead in the series, England needed eight runs from the last over but the 23-year-old Bumrah bowled with ice in his veins, conceding only two runs while claiming two wickets.
Both Ben Stokes and Joe Root made 38 but their contributions were not enough to get England across the line.
Root, though, was unlucky to be given out leg-before on the first ball of the eventful final over with replays suggesting an inside edge.
There was no argument, however, when Bumrah bowled Jos Buttler with his fourth delivery of that over.
Needing six runs of the last ball, Moeen Ali failed to connect with the Bumrah delivery.
Earlier put into bat, India posted a modest 144-8 with opener Lokesh Rahul (71) supplying nearly half of the total in an otherwise insipid batting display by the hosts.
Chris Jordan (3-22) drew first blood dismissing Virat Kohli for 21 in the fifth over but Rahul propped up the Indian innings with some crisp hitting, which included six boundaries and a couple of sixes.
The right-hander added 56 with Manish Pandey (30) to set the stage for a late assault which never came.
Clever bowling by Jordan, Stokes and Tymal Mills, backed up by England’s excellent fielding, stifled the Indian batsmen.
The pace trio, time and again, outfoxed the batsmen with well-disguised slower deliveries as boundaries dried up.
Jordan enhanced his death-bowling reputation with a tidy final over in which three Indian wickets tumbled with only five runs added to their score.
The target was modest but England wobbled early with Ashish Nehra (3-28) dismissing openers Sam Billings and Jason Roy off successive deliveries.
After Eoin Morgan’s exit, Root and Stokes looked poised to finish off the contest when Nehra injected fresh life into the match by dismissing Stokes.
Beaten in the preceding test and one-day series, England were very much in the contest but Root’s unfortunate dismissal and the lower order’s struggle against Bumrah meant England finished on 139-6.
The teams now move to Bangalore for the third and final Twenty20 International which marks the end of England’s tour.
(Reporting by Amlan Chakraborty in New Delhi; Editing by Ian Chadband)