Jade Dernbach’s record-equalling figures of 4-22 by an England bowler paved the way for victory in the one-off match after India were bowled out for 165.
England made heavy work of their chase, but with 10 needed from Vinay Kumar’s final over Samit Patel hit three successive fours to guide England home with three balls left.
It ensured Broad’s first success in his second game in charge with Eoin Morgan (49) leading the way after his 73-run stand with Ravi Bopara (31 off 36 balls) set up the successful chase.
England had initially looked set to concede a far bigger total after opener Ajinkya Rahane blazed a debut half-century.
The right-hander plundered 61 from 39 balls while fellow first-gamer, 38-year-old Rahul Dravid, added a quickfire 31 as the tourists passed 100 in 69 balls.
But England’s bowlers, led by Dernbach who equalled Paul Collingwood’s England-best figures taken against Sri Lanka in 2006, clawed their way back as India lost their final five wickets in 13 balls to set an achievable target.
England’s chase got off to a terrible start when Alex Hales, one of two debutants alongside Jos Buttler, was trapped lbw for a duck by Praveen Kumar from the second ball.
Despite the early loss, Craig Kieswetter and Kevin Pietersen played unhindered to combine for 58 from 37 balls.
Both struck the ball sweetly, with Pietersen surving a tough dropped chance by Parthiv Patel on 11, although they both fell in quick succession to cause some alarm.
Kieswetter offered a simple chance to Suresh Raina off Munaf Patel before Pietersen was stumped down the leg-side.
Morgan and Bopara then seemed to have the chase in hand when they combined in a stand of 73 from 49 balls.
The pair were happy to pick the gaps but also chose their moments to attack, blasting 17 off one over from part-timer Rohit Sharma.
They looked set fair to take England home before Morgan was caught by Sharma on 49, with the television umpire confirming his low catch.
England then hit a late lull and when Munaf Patel’s penultimate over cost just three runs they were suddenly left needing 10 from the final over.
But Patel held his nerve, slicing a pair of fours to third man before hitting over mid-off to sew up victory.
Afterwards, Broad insisted the players were never concerned about the final result, even heading into the final over.
“The guys felt we had it under control,” he said. “Morgan played pretty well, and then for Ravi and Samit to finish it off was pretty special.
“We’re delighted with the win, but there are things we can improve on.
“We lost an early wicket but the way Morgan and the guys batted was fantastic.To come back from the position we were in after 10 overs is fantastic.”
India captain Mahendra Singh Dhoni admitted a shortage of bowling options proved his side’s undoing, with injury robbing them of the likes of all-rounder Yuvraj Singh.
“You have to be at your best,” he said. “ We didn’t have a fifth bowler so we had to push harder.”