It seemed stand-in captain Morgan (49 not out) had left it too late, with three needed to win off the final ball from Ashok Dinda. But the man of the match kept his cool, striking a perfect blow back over the seamer's head to snatch the win and ensure England draw the two-match series.
"We're really pleased to get that victory. It was a world-class finish from a world-class finisher," said Jos Buttler, who shared in Morgan's celebrations at the end of the match as he finished unbeaten on 15.
"He was very confident and that rubbed off on me," Buttler added. "It's a great Christmas present. It was a world-class performance from everyone and everyone's delighted to get a win."
Michael Lumb – who shared an opening stand of 80 with Alex Hales – hit an important 50 from 30 deliveries as England successfully chased down India's 177 for eight.
Hales added 42 as the tourists made a bright reply, and Lumb admitted it was a special way to end England's 2012 programme.
"It was nice to get a few runs, but more importantly to win the game," said Lumb."Morgs at the end there and Jos, to get over the line. It was very special to go home with the win. Well done to the boys. Very special. Morgs has done it before, got us over the line."
Morgan enjoyed the close tussle and was full of praise for the performance of his players at the Wankhede Stadium.
"It was a hell of a game of cricket," he said. "It was a great pitch to be prepared for a Twenty20 game and both sides were in the game throughout the whole 40 overs, which was great to see."
Morgan won a crucial toss ahead of the match and was vindicated in his decision to put in the hosts to bat.
"The stats suggest, particularly at this stadium, that chasing predominantly wins, so it was part and parcel of what we looked at, and we backed ourselves to chase down or even set a good score."
England's hard-working players saved a number of runs with some great work in the field.
Morgan added: "It was outstanding. It's an area that we've looked at where we can be ahead of the game and save 10 or 15 runs each time, and today we were amazing."
Losing captain Mahendra Singh Dhoni was left to rue a slow start in both innings by his team, who top-scored through Virat Kohli's 38.
"What was vital was the start. I don't think we started really well, we were a fraction short," Dhoni said.
"As in the last game, we came back nicely, the bowlers started doing well again, but it was difficult. We gave up too many runs in the first six overs."