Australia, set what would have been a new Ashes fourth-innings record winning total of 412, were bowled out for 242 after tea as England went 1-0 up in the five-match series with more than a day to spare.
Stuart Broad led England’s attack with three for 39 as Ashes-holders Australia, who had been 97 for one, collapsed either side of lunch on the fourth day.
England made 430, with Joe Root top-scoring with 134 after being dropped on nought, before dismissing Australia for 308 to establish a first-innings lead of 122.
Their second innings 289 left Australia, who whitewashed England 5-0 on home soil during the last Ashes campaign in 2013/14, with a mammoth run chase.
No side have made more in the fourth innings to win an Ashes Test than Australia’s 404 for three at Headingley back in 1948 when Arthur Morris scored 182 and Donald Bradman, widely regarded as cricket’s greatest batsman, an unbeaten 173.
But the highest individual scores Australia, looking to win their first Ashes series in Britain in 14 years, managed on Saturday were No 8 Mitchell Johnson’s 77 and opener David Warner’s 52.
Root, appropriately, ended the match when he caught Josh Hazlewood at long-off from the bowling of off-spinner Moeen Ali.
The second Test at Lord’s starts on Thursday, with Australia fast bowler Mitchell Starc struggling to be fit for that match after suffering an ankle injury in Cardiff.
England 430 (J Root 134, M Ali 77, G Ballance 61, B Stokes 52; M Starc 5-114, J Hazlewood 3-83) and 289 (I Bell 60, J Root 60; N Lyon 4-75)
Australia 308 (C Rogers 95; J Anderson 3-43) and 242 (M Johnson 77, D Warner 52; S Broad 3-39, M Ali 3-59)
Result: England won by 169 runs
Series: England lead five-match series 1-0.