The visitors scored 316 for eight after being sent in to bat in the dead-rubber series, having conceded the first three games.
Emerging star Ben Stokes showed the way with a polished 70 off 84 balls, helped by Jos Buttler's late surge of 71 off 43. Opener Ian Bell kicked off with 55 from 52 deliveries on what looked a prime batting deck.
Under-pressure captain Alastair Cook made a crisp 44 from 43 balls at the top order, only to fall when a much-needed big score seemed likely for the embattled skipper.
For just about the first time on the tour, England got on top of paceman Mitchell Johnson, who went wicketless and conceded 72 runs from his 10 overs, although he was on the wrong end of two dropped catches.
The tourists, beaten in all eight of their previous encounters with Australia at Test and limited overs level on a nightmare tour, finally enjoyed some good fortune as the home team couldn't maintain their previous efficiency in the field.