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 today, 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.
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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Swann finished with innings figures of four for 78 for a match haul of nine for 122. Last man Ryan Harris was 16 not out after defying England for more than an hour.
No side has ever made more to win in a fourth innings of a Test than West Indies' 418 for seven against Australia at St John's in 2002/03.