England bowlers cashed in on the questionable shot selection of the Indian batsmen to restrict the hosts to 147 for seven, before Morgan led from front with a 51 off 38 balls to set up an emphatic win for his team.
Joe Root too played his part well to remain unbeaten with run-a-ball 46 as England overhauled the 148-run target with 11 balls to spare.
England, with their long batting line-up, were expected to chase down a target of 148 with little difficulty. Their destructive openers Sam Billings (22) and Jason Roy (19) provided a flying start with the visitors racing to 36 for no loss in three overs.
Jasprit Bumrah was the first to be at the receiving end after Ashish Nehra bowled a tidy first over. Billings was on fire from ball one as he smashed Bumrah for 20 runs in the second over. His stand out strike was the six over fine leg as he exposed all his stumps to dispatch the ball just over the ropes.
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The ploy worked immediately with leggie Yuvendra Chahal finding the stumps of both the dangerous openers in his first over, bringing the silent crowd back to life.
It was also a big moment for Jammu and Kashmir's sporting posterboy Parveez Rasool from the other end as he was making his T20 debut after playing his only international match way back in June 2014.
Root and Morgan then joined hands with England reaching 43 for two in the four over. The blistering start was provided by the openers and all Root and Morgan needed to do was forge decent partnership.
That is what they exactly did and without taking many risks, they took their team on the path of victory with a 83-run stand.
Morgan especially was brutal against the spinners and ended up smashing four sixes in his whirlwind knock. He could not be there till the end as he mistimed one to long-off, giving Rasool his first wicket in T20 Internationals.
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Earlier, loss of wickets at regular intervals did not allow India to accelerate even though MS Dhoni's 36 off 27 balls towards the end helped India get closer to 150.
The other contributions came from captain Virat Kohli (29 off 26), who opened the batting, and comeback man Suresh Raina (34 off 23).
Offie Moeen Ali was the stand out bowler for England, ending with figures of two for 21 in four overs. It was expectedly a packed house for the first ever T20 International at the Green Park.
Kohli, who had dropped hints of opening the batting in the pre-match press conference yesterday, kept everyone guess till the toss before coming out to open against Lokesh Rahul.
He made the move keeping the balance of the side in mind, allowing him to accommodate Raina at three and Manish Pandey at six.
India reached 47 for one in six overs with Rahul falling cheaply for the fourth time in as many innings. He was bounced out by pacer Chris Jordan to be caught at short-fine leg.
The second over bowled by Jordan saw couple of plays and misses from Kohli before the star Indian batsman spanked the English pacer over mid on for a four.
Local hero Raina was at the other end and with Rahul back in the dressing room in the fifth over, it was a golden chance to make a statement in his first match for India since the World T20 in March.
With Kohli watching from the non striker's end, Raina began in a fluent fashion hitting Liam Plunkett for two fours including a dead straight one with the ball bouncing just before the ropes.
Captan Kohli fell in the seventh over, trying to whip Ali over midwicket but the ball went straight into the hands of his counterpart Eoin Morgan. Raina departed in the 13th over and a ball after hitting the first six of the match.
Ben Stokes came with up a fitting response after getting whacked as he bowled a shuffling Raina behind the stumps with a well directed yorker.
Thanks to a late burst from Dhoni, India were able to post a fighting total. He smashed Jordan for consecutive fours in the final over, much to the delight of the vociferous crowd.