With England needing eight to win in Nagpur, Bumrah removed Joe Root and Jos Buttler, the former wrongly given lbw, and conceded only two runs.
Ben Stokes (38) looked to be taking England to their target of 145 before falling lbw to Ashish Nehra (3-28).
Opener KL Rahul earlier made 71 from 47 balls in India's 144-8.
That was three runs fewer than India managed in England's comfortable seven-wicket win in the first T20 game in Kanpur, but this chase was more challenging on a slower, lower surface.
And though the majority of England's attack impressed, their spinners were outbowled by India's, with pace bowlers Nehra and Bumrah making telling contributions at each end of the visitors' innings.
The deciding third match takes place in Bangalore on Wednesday.
England needed 24 from the last two overs, but were made favourites by the hitting of Buttler, who took 12 from Nehra's final three balls, complete with a bottom-handed fetch over the long-on fence.
It was left to jerky right-armer Bumrah to defend eight runs off the concluding six balls, which began with a slice of fortune that saw Root adjudged leg before, despite an inside edge, departing for a run-a-ball 38.
From there, Bumrah befuddled England with a full length and changes of pace - the tourists only laid bat on one of the last five deliveries, and even that was a Moeen Ali miscue for a single.
When a swiping Buttler was bowled, England's best hope went with him, and Moeen failed to make contact when a six was needed off the final ball.
England's chase was dealt an early blow by 37-year-old left-armer Nehra, who learned from the way the tourists bowled back-of-a-length to have openers Sam Billings and Jason Roy caught on the leg side from successive deliveries.
From 22-2, Root rebuilt in the company of captain Eoin Morgan, England happy to accumulate in the face of the accuracy of the India spinners as only three boundaries came in one 10-over period.
It took the power hitting of Stokes to ignite the chase, the left-hander making the most of a reprieve after being bowled by no-ball by spinner Amit Mishra, from the first delivery he faced.
Muscling the ball to long-on for two fours and two sixes, Stokes looked to be in control of the pursuit before being pinned in front by the returning Nehra.
When he departed, England required 28 from 19 deliveries. As Bumrah and Nehra excelled, they managed only 19, and 10 of those came from two Buttler blows.
Whereas India's spinners used the conditions to squeeze England in the middle overs, the home batsmen pounced on slow bowlers Adil Rashid and Liam Dawson.
Rashid, who did not bowl in Kanpur, and Dawson, in the side for paceman Liam Plunkett, conceded a combined 44 runs from their five overs.
Rahul, who averages 64.50 in his seven T20 internationals, took each of them for a leg-side maximum to go with the sweet cover drives he played off the fast bowlers.
Along with the rest of the middle order, Rahul was tied down by Moeen, the off-spinner returning excellent figures of 1-20 to give him a collective 3-41 from eight overs in the series, with only one boundary conceded.
Rahul eventually lofted Chris Jordan to deep mid-wicket, the second of three wickets for the Sussex paceman who also could have had Virat Kohli lbw for seven.
The reprieved Kohli went on to punish Tymal Mills, who recovered as part of a controlled display by England's pace attack, particularly at the death. But by then, the damage had been done.
England captain Eoin Morgan: "We were in the driving seat, right up until the last over. We didn't start the over well - a decision didn't go our way and losing an 'in' batsman like Joe Root hurt us. We bowled really well again.
"We have struggled to string performances together with the ball, but we have been right on the money in the last two games. The chase never got too far away from us. Joe Root played really well, as did Ben Stokes. It's unbelievably disappointing to lose this game but we have to take it on the chin."
India captain Virat Kohli: "Bumrah was asking me after every ball what he should do and I said 'just play your game and if it goes for a six we still wake up tomorrow and everything is OK'."
Man of the match Jasprit Bumrah: "I tried to assess the wicket, to see what has happened in the first innings and what has worked for other bowlers. On this, it was a big ground and a slow wicket so back-of-a-length and slower balls were best.
"The confidence from the captain helps. He has given me the freedom to execute the plans. I am happy that we are on the winning end."
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Jasprit Bumrah bowled a brilliant final over to give India a thrilling five-run win over England in the second Twenty20 international to level the series.