Australia, set a target of 183 for victory, were on course while skipper Steven Smith was making a Twenty20 best score of 90.
But Australia lost five wickets for 16 runs in 14 balls, with three batsmen dismissed in the final over of the innings, bowled by Ben Stokes, as they finished on 177 for eight.
Earlier, England made 182 for five after being sent into bat by Smith.
Eoin Morgan, the England captain, top-scored with 74 and put on 135 for the third wicket with Moeen Ali (72 not out).