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Victory, achieved with more than a day to spare, saw England take the five-match series 2-1.
Australia, set what would have been a record fourth innings victory total of 546, were bowled out for 348 with Michael Hussey last man out for 121, when he was caught close in by Alastair Cook off spinner Graeme Swann.
From 327 for five, the end came swiftly with four wickets going down for 16 runs in 32 balls, with fast bowler Stephen Harmison taking three for seven in 13 himself.
It was a remarkable turnaround for England, who'd lost the fourth Test by an innings and 80 runs at Headingley, and had only clung on for a draw in the series opener at Cardiff by a single wicket.
Australia had been making steady progress at 217 for two before captain and star batsman Ricky Ponting was run out by Andrew Flintoff's direct hit in the all-rounder's last match before his Test retirement.
Five balls later Michael Clarke was also run out, for nought, before England's progress was checked by a sixth-wicket stand of 91 between Hussey and Brad Haddin.
But just when England fans were starting to fret again, Haddin, on 34, went down the pitch to Swann, who took four for 120 in 40.2 overs, and hoisted a mistimed drive to a jubilant England captain Andrew Strauss at mid-wicket.
And 327 for six became 327 for seven when Australia lost another two wickets in quick succession.
Mitchell Johnson went for a seven-ball nought, brilliantly caught by diving second slip Paul Collingwood off his Durham team-mate Harmison.
Peter Siddle then got a leading edge off Harmison and Flintoff held a simple catch at mid-off.
Harmison made it two wickets in two balls when Stuart Clark was snapped up at short leg by Cook.
Ben Hilfenhaus survived the hat-trick but the end was nigh.
England had taken the new ball after tea with Australia 278 for five.
But Hussey's two off Stuart Broad saw him to a first Test century in 29 innings off 219 balls with 11 boundaries.
Flintoff and Strauss had stunned Australia with two run-outs, the first they'd suffered this series, before tea to dismiss Ponting and Clarke.
Ponting and Hussey had frustrated England with a third-wicket stand of 127 that raised hopes of an improbable victory.
But Flintoff struck in unlikely fashion.
Hussey called Ponting for a single off Harmison but Flintoff, running round from mid-on, threw down the stumps at the striker's end and Ponting was short of his ground by a foot.
Ponting, in what could be his final Ashes Test in England, had to go for 66, having stroked 10 stylish fours in his 103-ball stay after he and left-hander Hussey had revived Australia from the depths of 90 for two.
Vice-captain Clarke fell in even more extraordinary fashion.
Australia's leading batsman this series, clipped a ball from Swann that deflected off the boot of Alastair Cook at short leg to leg-slip Strauss, who hit the stumps with a sharp under-arm throw.
This was a much tighter call for third umpire Peter Hartley but he eventually ruled the bail was in the air with Clarke's bat still to be grounded.
Clarke was out for nought and Australia, who'd lost two wickets for three runs in six balls, were now 220 for four.
And they could have been five down had not Collingwood dropped Hussey, on 55, off Swann.
But Australia were 236 for five when Marcus North, trying to sweep Swann, was smartly stumped by wicket-keeper Matt Prior.
Australia resumed Sunday on 80 without loss.
Left-hander Simon Katich had only added one to his overnight 42 when, padding up, he was plumb lbw to Swann.
Three balls later, fellow opener Shane Watson (40) was lbw to man-of-the-match Stuart Broad.
England consolidated Saturday the advantage gained from dismissing Australia for just 160, with Broad taking five for 37, in a second innings 373 for nine declared that featured a debut century from Jonathan Trott, who made 119.
Strauss's declaration left the visitors needing to break the record for a successful Test fourth innings run-chase, of 418 for seven set by West Indies against Australia in Antigua in 2002/03, to win this match.
England victory's left Ponting with the unwanted record of becoming only the second Australia captain, since Billy Murdoch in 1890, to be in charge of two losing Ashes tours of England.
And it also knocked Australia off top spot in the Test rankings.