Morgan, who had scored only 198 runs in 10 one-day international innings since returning from injury in March, smashed 71 off 31 balls including seven sixes and three fours, as England posted a daunting 180 for seven wickets in their 20 overs.
In reply, another batsman out of sorts this summer - Virat Kohli - led the India charge but his 66 off 41 balls and 27 off 18 by India captain Mahendra Singh Dhoni at the death could not clinch victory for India who finished on 177 for five.
Ajinkya Rahane took a T20 record-equalling four catches, including a stunning running, diving effort to dismiss Alex Hales (40) but India's bowling took a battering late on, conceding 81 runs off the last five overs as Morgan and Ravi Bopara ran riot.
Bopara added 21 off 9 balls with three fours and a six while Mohammed Shami was the pick of the Indian attack, taking three for 38.
"It was awesome, games like this are great for a team like ourselves as we're learning, and we've held our own against some of the best in the world - [Harry] Gurney, [Chris] Woakes and [Steven] Finn came up trumps," Morgan told the BBC.
"You've very little margin for error with someone like Dhoni at the crease. To come out with a win today, the guys should be proud of themselves.
"It was nice to get some runs today, I was striking it really well and targeting the short boundaries. People will leave the ground, and leave their television sets, with a smile on their face."
England smashed 17 runs off the first over with 24-year-old debutant Jason Roy and Hales starting aggressively. But South Africa-born Roy was first to go for eight, chipping to Rahane at cover off Shami.
Two balls later Moeen Ali fell for a duck - caught again by Rahane off Mohit Sharma - then Hales and Joe Root steadied the ship, putting on 48 before Hales fell.
Root, a century-maker in the final one-day international between the sides, was well caught by Ambati Rayudu for 26 as Morgan led the final onslaught.
India's reply got off to a poor start, Rahane lasting four balls, bowled by Ali, before Kohli and Shikhar Dhawan led the India charge.
Kohli and Dharwan put on 79 in nine overs before Chris Woakes bowled Dharwan.
Kohli survived a difficult catch to Harry Gurney when on 65 but fell one run later, tucked up by Steve Finn and caught in the deep by Hales.
India were always up with the scoring rate but a mix-up between Dhoni and Ravindra Jadaja seemed to have ended their hopes before Dhoni set about Woakes in the final over with India still 17 runs short.
Dhoni smashed 12 off the first four balls before Woakes managed to subdue the India skipper and Englnd scraped home before a packed crowd.
(Reporting By Tony Goodson; editing by Justin Palmer)
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