Man-of-the-match Morgan finished unbeaten on 103, steering England to their target of 268 with four overs to spare after they had lost four early wickets.
Andrew Strauss fell for 10, Kevin Pietersen for 29, Paul Collingwood for 11 and Craig Kieswetter for 38 to leave the hosts in trouble on 98 for four.
But Morgan received good support from Luke Wright who made a solid 36 and Tim Bresnan whose knock of 27 took the hosts to the brink of victory.
Earlier, Michael Clarke made a fluent unbeaten 87 to lead Australia to a competitive total of 267 for seven.
Australia won the toss and openers Shane Watson and Tim Paine added 52 for the first wicket in perfect batting conditions at the Rose Bowl in Hampshire.
Watson was caught and bowled by Stuart Broad for 32, Paine was bowled by Wright for 26, and when captain Ricky Ponting was caught on the long-leg boundary by Broad off Wright for 21, the Australians were struggling at 86 for three.
Cameron White (10) and Mike Hussey (28) also fell cheaply before Clarke, who hit seven fours, was joined by all-rounder James Hopes and the pair put on 57 for the sixth wicket.
The second match in the five-game series is in Cardiff on Thursday.
Scorecard:
Australia innings
S. Watson c and b Broad 32
T. Paine b Wright 26
R. Ponting c Broad b Wright 21
M. Clarke not out 87
C. White b Anderson 10
M. Hussey c Kieswetter b Yardy 28
J. Hopes c Bresnan b Anderson 34
N. Hauritz c Strauss b Broad 22
R. Harris not out 0
Extras (lb-2, w-4, nb-1) 7
Total (7 wickets; 50 overs) 267
Fall of wickets: 1-51, 2-66, 3-86, 4-98, 5-168, 6-225, 7-266.
Bowling: J. Anderson 9-1-43-2, T. Bresnan 8-1-49-0, S. Broad 8-0-54-2, L. Wright 7-1-34-2, M. Yardy 10-2-41-1, G. Swann 8-0-44-0.
England innings
A. Strauss c Paine b Harris 10
C. Kieswetter b Hazlewood 38
K. Pietersen c Ponting b Watson 29
P. Collingwood c Hopes b Watson 11
E. Morgan not out 103
L. Wright lbw b Harris 36
T. Bresnan b Harris 27
M. Yardy not out 1
Extras (lb-8, w-4, nb-1) 13
Total (6 wickets; 46 overs) 268
Fall of wickets: 1-16, 2-75, 3-81, 4-97, 5-192, 6-263
Bowling: D. Bollinger 9-0-48-0, R. Harris 9-2-42-3, J. Hazlewood 7-0-41-1, S. Watson 8-1-55-2, J. Hopes 5-0-30-0, N. Hauritz 8-0-44-0.
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