The hosts were all out for 567 early on day four, during a morning session in which England's inspirational leader Ben Stokes limped from the SCG with a groin injury.
After passing three figures on Tuesday, Steve Smith was eventually out for 138 on Wednesday morning, while Beau Webster finished unbeaten on 71.
With a little over five sessions left in this match and signs of inconsistent bounce, a draw is still very much a possibility.
But England look a side thinking about their impending flights home, having dropped six catches in Australia's innings and appearing bereft of idea.
On the past three tours of Australia, England have surrendered meekly in their final innings, averaging just 156 and failing to last more than 40 overs in two of them.
If that trend is followed again, this Test could be over as soon as Wednesday, with an innings victory on the cards for Australia.
Whether the tourists will have Stokes available to bat is also unclear.
With other quicks struggling and the recalled Matthew Potts going for 0-141 off 25 overs, Stokes over-bowled himself before injuring his groin on Wednesday.
Any further English pain was at least put on hold when Smith edged Josh Tongue behind two overs later, falling to the seamer for the fourth time in three Tests against him.
Webster's runs, meanwhile, will add to the calls for him to replace Cameron Green as Australia's first-choice allrounder.
After Green threw away his wicket when caught hooking in the deep on 37 on Tuesday, Webster looked at complete ease through his whole innings.
He hit three boundaries on the pull shot and flicked nicely off his pads on the way to his fifth half-century in eight Tests.
After making his debut in Sydney last summer, Webster now has 449 Test runs at an average of 40.81, to go with his eight wickets at 23.25 with the ball.
Green, in contrast, has had a difficult Ashes series, with 149 runs at 21.28 and no scores above 50.
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