Victory saw Ashes-holders England go 2-0 up in the five-match series.
Only once in Ashes history have a side come from 2-0 down to win a series when Australia, inspired by Don Bradman, cricket's greatest batsman, triumphed 3-2 on home soil in 1936/37.
Australia, set a mammoth 583 to win Sunday, were bowled out for 235 on the fourth day after Joe Root, who earlier made 180 in England's second innings 349 for seven declared, sparked a collapse with his occasional off-spin.
It looked as if Australia's last-wicket duo might see out the day, with England claiming the extra half hour.
There were just three balls left in the day's play when off-spinner Graeme Swann, who took five for 44 in Australia's meagre first innings 128, ended James Pattinson's two hours of resistance by having him lbw for 35.
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Heavenly intervention via Usman Khawaja failed to help Australia.