England lose Crawley as they battle to save final Ashes Test
Ben Duckett was not out 40 and Jacob Bethell on 28 at the Sydney Cricket Ground after the hosts were dismissed for 567 on the back of Travis Head's 163 and 138 from skipper Steve Smith.
That left the tourists chasing a 183 deficit after their first innings 384, fighting to salvage the game with a question mark over Ben Stokes's ability to bat.
The England captain limped off clutching what appeared to be his groin while bowling the third over of the day and did not reappear.
Australia lead the series 3-1 and have already retained the Ashes, with England desperate for another morale-boosting win after victory in the previous Test at Melbourne.
They needed a solid start to the chase, but chief tormentor Starc once again conjured up a first-over breakthrough -- his 29th wicket for the series.
The pace spearhead delivered a beautiful inswinger that Crawley misjudged and left, with the ball smacking his pads and he was out lbw for one.
Duckett and Bethell, who both need a big score, built a 76-run partnership though Duckett was lucky to survive, dropped on 38 just before lunch by Cameron Green diving in front of Smith at first slip.
Australia resumed at 518-7, with Smith on 129 after his 13th Ashes century -- more than any other player except the legendary Don Bradman -- and Beau Webster on 42.
Smith flogged Stokes through the covers for a boundary in his first over, while Webster dished out similar treatment to Brydon Carse.
But the Australian skipper did not last long, nicking a fuller length delivery from Josh Tongue to wicketkeeper Jamie Smith.
All-rounder Webster reached a fifth half-century in his eighth Test with a single and put on 20 with Starc before the big paceman was bowled by Tongue for five.
Scott Boland suffered a golden duck, edging Will Jacks to Harry Brook in the slips to leave Webster unbeaten on 71.
Tongue ended with 3-97 and Carse 3-130.