Sharp bowling by Josh Tongue forced the hosts to make a meal of it, but finally chased down the 160 target with Alex Carey and Cameron Green at the crease after losing a flurry of wickets.
Earlier, England resumed on 302/8 with centurion Jacob Bethell and Matthew Potts, their lead 119. Australia waited for the new ball and Bethell cashed in, reaching his 150. The visitors' lead had then swelled to 145 and hope was building.
But Mitchell Starc, the man who's been crushing England's hopes all summer Down Under, got the breakthrough. Bethell tickled one to Carey for his umpteenth catch of the series after being cramped on a pull shot.
Tailenders Potts and Tongue scratched around, but the indomitable Starc got his man 14 runs later when the latter chipped it to cover.
Openers Travis Head and Jake Weatherald were positive up front and raced to 57/0 off the opening ten overs. Keen to put England to the sword once again, Head (29) overplayed and skied Tongue to midwicket. Despite a good start, Weatherald (34) fell in the last over before lunch, top-edging Tongue to fine leg.
Australia captain Steve Smith and Marnus Labuschagne were brisk after the break, but a ripper from spinner Will Jacks castled a perplexed Smith, leaving the hosts 92/3.
The stage was set for the retiring Usman Khawaja to give himself a fairytale ending to his career. The 39-year-old struck one final boundary, then dragged one onto his stumps off that man Tongue. Khawaja (6) received an incredible ovation from the 25,000-strong Sydney faithful and basked in its glory, kissing the turf on his way off the hallowed ground.
But the fan furore quickly died down in the next over when a mix-up in the middle led to Labuschagne (37) being run out. Having lost five wickets for just 59 runs, Australia were wobbling.
Carey and Green guided their beloved Baggy Greens home for the last 39 runs, with Carey fittingly hitting the winning runs to cap off a brilliant series where he took a remarkable 27 catches and scored 323 runs.
Starc was awarded the Player of the Series for his 31 wickets to go with 156 important runs down the order.
England, on the other hand, did a lap of the SCG to thank the Barmy Army, who were perhaps their most consistent performer of the series.
Australia defied pre-series claims that their side was too old by overpowering England 4-1 in the Ashes, with a veteran-heavy team turning "Dad's Army" jibes into a commanding series victory.
England white-ball captain Harry Brook has been fined by the England and Wales Cricket Board following an altercation with a bouncer in New Zealand in October.
Retiring Australia batter Usman Khawaja has admitted that emotion got the better of him during his 88th and final Test, despite a five-wicket victory and a 4-1 Ashes series triumph.
England captain Ben Stokes heaped praise on his Australian opposition after losing the final Ashes Test by five wickets in Sydney on Thursday.
