In an absolute thriller, replacement Max Jorgensen went over for a glorious score in the 83rd minute on Saturday with the final attack of a pulsating match.
It gave Joe Schmidt's side a thoroughly deserved 42-37 win, Australia's first victory over England at Twickenham since their World Cup triumph nine years ago.
It concluded a sensational match in which the hosts looked as if they had snatched victory when Maro Itoje went over for a converted try in the 79th minute, just four minutes after Andrew Kellaway appeared to have won it for the Aussies with a break from his own half.
Aukuso-Suaalii's eagerly awaited debut in his first rugby union match since he was a schoolboy five years ago featured a few fleeting moments of brilliance from the young $5 million cross-code recruit as he helped set up one of Australia's five tries in the first half.
He was even - rather generously - awarded the man of the match award for his fine performance.
Tom Wright, captain Harry Wilson and lock Jeremy Williams were the Wallabies' other scorers as they ended a three-defeat sequence in encouraging style with some brilliant attacking.
England had been rampant from the outset, Suaalii thrown straight into defensive duty as the hosts crashed over for a score in the right corner, following a sharp move initiated by Smith's neat midfield grubber, after just five minutes with flanker Chandler Cunningham-South barging over in the corner.
With Smith running the show, Harlequins No.6 Cunningham-South then ploughed over again, put in after a short blasting tap-and-run by Ellis Genge.Â
Amid some helter-skelter early attacking from both teams, Smith's 19th minute penalty gave England a 15-3 lead after just the first quarter, with things looking grim for the visitors.
Enter Suaalii. Noah Lolesio had narrowed the margin with a penalty and when Australia attacked down the left, a beautifully timed one-handed pop pass over the top from the 21-year-old as two tacklers came bearing down on him, put Wright away for a superb try.
The Wallabies were starting to look sharp and threatening on every attack, and Tate McDermott, on as a blood bin replacement for Jake Gordon, quickly dazzled when he burrowed through a gap and sent captain Harry Wilson in for a score he'll never forget.
When the excellent Lolesio slipped over another penalty just before the break, it seemed almost unthinkable that Australia went in at the break leading 20-18 after their wretched start.
The Wallabies perhaps began dreaming once Western Force lock Williams ploughed down in the corner after a short-side attack, the TMO confirming the on-field ruling after a lengthy review that he had grounded the ball fractionally before his left foot slid into touch.
Lolesio put the Wallabies 10 points clear after a stirring counter-attack but another delicious grubber from Smith was latched on to by replacement Ollie Sleightholme for a brilliant riposte.
And when the Northampton man sped over for another in the 68th minute, with Suaalii left exposed in defence, it looked as if the Australians were finished.
But Kellaway snaffled up a misplaced pass to scoot over from his own half with five minutes left, only for England to forge back one last time amid huge cheers from the 82,000 home crowd as Itoje scored, with the brilliant Smith adding the extras to give England a two-point lead with just seconds left.
But Australia worked one last chance to score a try for the ages, a brilliant movement slipping swiftly through three pairs of hands, with Len Ikitau proving the last remarkable one-handed offload that sent Jorgensen powering up the left touchline and skydiving over the line.