Morgan made 74 and shared a third-wicket stand of 135 with fellow left-hander Moeen Ali (72 not out) in a total of 182 for five after England, sent into bat, had slumped to 18 for two.
His innings was a vindication of the month's break Morgan had from cricket after being rested by Middlesex -- whose director of cricket, Angus Fraser, is also an England selector -- following a poor run of domestic form.
Australia, however, were on course for victory while skipper Steven Smith was making a Twenty20 international best 90.
Australia needed 12 to win off the last over, bowled by all-rounder Ben Stokes.
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However, they lost three wickets, two to run outs, with Stokes also having Nathan Coulter-Nile out for a duck when Adil Rashid held a skyer at short third man.
Stokes conceded just six runs off the over and finished with one for 29 in his maximum four-over stint.
Earlier, England lost openers Alex Hales (three) and Jason Roy (11) in four balls from paceman Pat Cummins to be 18 for two in four overs.
England took 19 runs off Cameron Boyce's lone over of the innings.
It meant the leg-spinner, who did not feature in the Ashes and has not been included for the five one-day internationals that follow Monday's match, was left contemplating a round trip of some 20,000 miles to bowl six balls in international cricket this tour.
Morgan also hammered Cummins for two sixes before he was well-caught on the long-on boundary by Watson off paceman Coulter-Nile.
Cummins, however, came back well to finish with two for 25. Man-of-the-match Ali, deprived of the strike late on, faced 46 balls and struck six fours and two sixes.