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Australia had two tries disallowed in close calls by Irish Television Match Official Simon McDowell, who also allowed England's early try by Elliot Daly after a lengthy review.
Australia also had two players sin-binned in the first half as their seven-match unbeaten run ended with a fifth successive defeat against England.
The key moment of a game England coach Eddie Jones described as an "arm-wrestle" came 12 minutes from the end when the Wallabies were on top and thought they had levelled at 13-13 when Marika Koroibete forced his way over.
Instead it was ruled out for obstruction and England took full advantage as, in a crazy finish, Jonathan Joseph, Jonny May and Danny Care scored to turn what had been a knife-edge game into their biggest victory over their old rivals.
"There was not much in it but we had the bounce of the ball and we always back ourselves to play for 80 minutes," Jones said.
"Our finishers did a great job but tactically we played smart in the end – the conditions were tough and the ball was wet but I always thought we would finish stronger than them."
Australia captain Michael Hooper was at the heart of things, having a try disallowed and then being sent to the sin-bin.
"It got away from us at the end. There were a couple of decisions that didn’t go our way," he said. "It’s hard to judge when you’re on the field, but I thought they were both tries.”
England showed more intent than in last week's "grindathon" victory over Argentina but on a cold day with a greasy ball there was a lot of tactical kicking and not much ball in hand.
England edged ahead with two Owen Farrell penalties and Australia reached halftime with Hooper and fullback Kurtley Beale in the sin-bin to leave them down to 13 men and trailing 6-0.
A long-range penalty by Reece Hodge closed the gap but one of two key moments then occurred in the 54th minute. A brilliant tackle-bursting run and offload by Samu Kerevi looked set to put Tevita Kuridrani through but his centre partner knocked on.
England regathered and Ben Youngs’ long kick deceived Beale by hugging the touchline but slicing in. As Beale relaxed, Daly was alive to it, hacked forward and scored the key try, which, after a lengthy TMO discussion, Farrell converted.
Instead of Australia leading 10-6 they were suddenly 13-3 down with 25 minutes to go.
The visitors still had more of the game and starved England of the ball, reducing the gap with a Foley penalty. They then cut through more shaky England defending to send Koroibete over – only for the TMO to again rule against them, disallowing the try for obstruction.
Instead of being level, Australia were suddenly on the ropes as England finally got behind them.
Replacement Care chipped through for Joseph to chase and slide over, May showed remarkable determination to force his way over after another Care kick and the scrumhalf completed a memorable cameo by scoring the final try.
Australia coach Michael Cheika was not a happy man in his post-match news conference but accepted that his team had failed to take their chances.
"On the TMO, my only issue is the number of replays to look at one incident against the number on another," he said.
"But we have a culture of no excuses and we had enough opportunities, we just didn't take them."