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.
Earlier, Michael Clarke [ Images ] made a fluent unbeaten 87 to lead Australia to a competitive total of 267 for seven.
Australia won the toss and openers Shane Watson [ Images ] 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 [ Images ] for 32, Paine was bowled by Wright for 26, and when captain Ricky Ponting [ Images ] was caught on the long-leg boundary by Broad off Wright for 21, the Australians were struggling at 86 for three.
Cameron White [ Images ] (10) and Mike Hussey [ Images ] (28) also fell cheaply before Clarke, who hit seven fours, was joined by all-rounder James Hopes [ Images ] and the pair put on 57 for the sixth wicket.
The second match in the five-game series is in Cardiff on Thursday.
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.
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.