Australia, chasing a mammoth 384 to win, collapsed from 264-3 to 334 all out, with England's retiring paceman Stuart Broad taking the final wicket.
England did the bulk of the damage when, after a rain break of more than two hours, they took four Australia wickets for 11 runs in 19 balls.
Off-spinner Moeen Ali (3-76) and all-rounder Chris Woakes (4-50) led the way before Broad, England's all-time leading Ashes bowler polished off the tail.
"It was absolutely wonderful," Broad told Sky Sports. "I thought Chris Woakes and Moeen Ali set the tone absolutely unbelievably.
"Woakesy picked up a couple of wickets, particularly Steve Smith, who has just been a wonderful player to play against all these years.
"Once we got a couple we really started to believe. The crowd were unbelievable. It was so loud and we just jumped on the back of that. To contribute to the team with two wickets is very special.
"When you make that decision you wonder what your last ball will be so to take a wicket to win an Ashes Test match is pretty cool."
Australia, as the holders, had already retained the Ashes but they remain without an Ashes series win in England since 2001.
When play resumed after the rain delay, the tourists were 238-3, needing 146 more runs for victory.
Steve Smith, dropped at leg gully by England captain Ben Stokes just before the interval, was 40 not out and left-hander Travis Head 31 not out.
MOEEN TRIGGERS COLLAPSE
The pair had extended their fourth-wicket stand to 95 when off-spinner Moeen, in what could be his last Test appearance, turned a ball out of the rough.
Head, on 43, tried to drive, with Joe Root holding a routine catch at first slip.
Australia then lost their star batsman when Smith fell for 54, edging an excellent full-length delivery from paceman Woakes to Zak Crawley at second slip.
The tourists slipped to 274-6 when Moeen had all-rounder Mitchell Marsh brilliantly caught by diving wicketkeeper Jonny Bairstow before Woakes and Crawley combined to dismiss Mitchell Starc for a duck.
Captain Pat Cummins was next to go, caught by Stokes off Moeen for nine.
But with Australia needing 55 to win, Broad, who on Saturday made a shock announcement that he would retire after the match, produced a brilliant delivery to take tailender Todd Murphy's outside edge, with Bairstow holding another fine diving catch to leave Australia on the brink of defeat at 329-9.
Broad then had Carey edging behind only for Crawley to drop a tough low chance at second slip before the 37-year-old ended his career in fairytale fashion by having the wicketkeeper caught behind.
Earlier, all-rounder Woakes, often one of England's unsung heroes, reduced Australia to 141-2 by removing openers David Warner (60) and Usman Khawaja (72) in a stunning burst of two wickets for one run in seven balls.
Express quick Mark Wood then had Marnus Labuschagne caught in the slips for 13 to leave Australia 169-3.
"We were going nicely there," Starc said. "Credit to England, they put some good balls in the right areas and we nicked them or weren't good enough.
"This has been the closest, most exciting series I've been a part of," he added. "All the test matches have been really special. It's the right result."
