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Four years after the Wallabies sent England packing out of their own tournament, Eddie Jones's side gained a measure of revenge to extend their dominance over the Australians to seven successive matches.
Australia played their part in an engrossing contest but in the end were well beaten by a team who defended stoutly, took their chances ruthlessly and had a flyhalf in Owen Farrell who managed the game maturely and kicked flawlessly for 20 points.
England have now three out of three World Cup quarter-finals against the twice world champions following their upset wins of 1995 and 2007 and will move on with confidence to the clash with the All Blacks or Ireland, who play later on Saturday.
Australia started well but were unable to convert early possession into points until Christian Lealiifano opened the scoring with a penalty after 11 minutes.
The lead would only last seven minutes before the first of May's double strike, England giving the Wallabies an object lesson in how to make pressure pay when the winger went over on the overlap after a beautifully delayed pass from flanker Tom Curry.
Australia went straight back on the attack but a David Pocock pop-pass went through the hands of Lealiifano and Henry Slade swooped onto the ball, the centre racing 40 metres before putting May into the left corner again with a neat grubber kick.
The Wallabies were mostly avoiding kicking out of defence but made good ground with ball in hand and again worked their way deep into England territory only to spill the ball in the tackle.
The referee was playing advantage, however, so Lealiifano was able to cut the deficit to 14-6 from the kicking tee, with another penalty apiece before halftime sending England into the break with an eight-point advantage.
Three minutes into the second half and it was one point, a lofted pass from Reece Hodge setting centre Jordan Petaia free on the left, the teenager finding Marika Koroibete who turned May inside and out before touching down.
Another England double whammy almost immediately put the Australians on the back foot again, however.
Farrell first provided a sublime flat pass to put prop Kyle Sinckler in for his first test try and then, after his pack had monstered Australia at the scrum, the flyhalf stepped up to extend the lead to 27-16.
Australia knew they needed to score next but, when offered the opportunity with a penalty in front of the posts just before the hour mark, Michael Hooper elected to go for a scrum.
England's eight held firm, the defence held firm, and soon they were back in the Australia half hammering away at the try line and earning a penalty that Farrell slotted to take the lead to 14 points.
The 2003 champions were now in full control and Farrell slotted a fourth penalty after Australia collapsed a rolling maul.
The Wallabies were forced to throw caution to the wind and Anthony Watson scored England's fourth try from an intercept four minutes from time to seal a dominant victory.