Opener Michael Carberry, who had made 43 not out despite his bat snapping in two at one stage, and Gary Ballance (7) were at the crease at the break still trailing Australia by 360 runs with seven sessions remaining in the match.
England captain Alastair Cook had concluded his contribution to the series when he was caught behind for seven, Ian Bell was dismissed for 16 and Kevin Pietersen departed for six.
Australia opener Chris Rogers earlier recorded his highest Test score with a knock of 119 before being caught and bowled by debutant leg spinner Scott Borthwick in the second over after lunch.
England mopped up the tail to dismiss the hosts for 276 in their second innings when England paceman Boyd Rankin removed Peter Siddle caught behind for four to claim his first Test wicket.
The tourists were always up against it given the highest successful fourth innings run chase at the Sydney Cricket Ground was the 288 Australia achieved to beat South Africa in 2005-6, and the start to their innings did not help.
In the second over, Cook's miserable tour continued when he prodded at a Mitchell Johnson delivery and caught an edge which Brad Haddin scooped up behind the stumps.
The England skipper's seven runs gave him 246 at an average of 24.6 for the series, a poor return compared to the 766 runs at 127 he accumulated on his last trip to Australia in 2010-11.
Bell, the difference between the two sides in England's 3-0 triumph in the first Ashes series of the year, faced 19 balls before he too headed back to the pavilion after cutting the ball straight to Dave Warner at gully off Ryan Harris.
England were now 37 for two and looking to their last experienced batsman, Kevin Pietersen, to give the innings some stability.
George Bailey put paid to those hopes when he took a brilliant catch after a Harris delivery had taken an inside edge and looped over his head off Pietersen's pad.
While Australia's bowlers continue to make life difficult for England's batsmen on a green-tinged pitch, the tourists' bowlers had failed to make early inroads into the Australian order after reducing them to 140 for four overnight.
Although he got some help from George Bailey (46) and Haddin (28), Rogers will rightly take the plaudits for bulking up the Australia score after his second century in successive matches and a third in the 10 Ashes Tests this year.
Resuming on 73 not out, the 36-year-old eased towards three figures with some fluency but also the caution that was once the hallmark of a Test opening batsman.
Reaching 99 with a single which also brought up the 100 partnership with Bailey, he passed the milestone in style with a splendid cut past point off Pietersen's occasional spin for his 14th four.
The Sydney-born lefthander removed his helmet and raised his bat to acknowledge the warm applause from a crowd bathed in sunshine and mostly dressed in pink in honour of fast bowling great Glenn McGrath's breast cancer charity.
Bailey departed in the next over and Haddin passed 3,000 Test runs before dragging a Borthwick delivery on to his stumps.
Australia were now attacking the bowling with the freedom of a side who knew they already had enough runs on the board and 22 runs were added by the tailenders even after the departure of Rogers.
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