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Warner and Bancroft's 173-run stand broke an 87-year-old record for the highest ever unbeaten opening stand in a successful chase in Test cricket.
Warner was unbeaten when Bancroft brought up the winning runs.
It was Warner's ninth Ashes half-century and his 25th in all Tests, while for Bancroft, it was a maiden Test fifty.
With support from Pat Cummins, who made 42 in an eighth-wicket stand of 66, Smith made sure Australia eked out a slender but important 26-run first-inning lead.
England seemed well placed to make a match of it despite losing two early wickets, but the dismissal of captain Joe Root, trapped leg before by Josh Hazlewood for 51 shortly before lunch on day four, led to a remarkable collapse.
The last six wickets went down for just 82 runs as England were bowled out for 195 in 71.4 overs.
The teams now head to Adelaide for the first pink-ball, day-night Ashes Test starting on December 2.