The touring side, chasing 311 for victory, battled into a strong position on 161 for three before losing captain Michael Clarke, Steve Smith and Phil Hughes in quick succession to tilt an extraordinary match back England's way.
Brad Haddin, on 11, and Ashton Agar, one, will resume in the morning with Australia still needing 137 to win and the home team requiring four wickets.
Earlier, Ian Bell made 109 and Stuart Broad 65, stretching their important seventh-wicket partnership to 138 as England made 375 in their second innings.
Bell and Broad, resuming on 326 for six, quickly reached the individual milestones their tenacious stand deserved.
Left-arm fast bowler Mitchell Starc got Australia's day off to a bad start with a wild beamer which flew to the boundary and Broad then slashed James Pattinson for four to get to his half-century.
Bell pushed Starc for a single to post his 18th test century, a marathon effort of intense concentration lasting more than six hours.
The 31-year-old Bell, often criticised for failing to deliver under pressure, leapt up and punched the air after completing his run before raising his bat to all sides of the ground.
Broad's fine knock ended when he edged paceman Pattinson through to wicketkeeper Haddin.
The England fast bowler walked straight off to ironic cheers after a controversial incident on Friday when he edged Agar to slip and stayed at the wicket having being given not out by the umpire.
Bell nicked a good delivery from Starc to Haddin and left the field to a standing ovation after hitting 15 fours.
Australia wrapped up the England tail quickly. Graeme Swann, on nine, edged the persevering Peter Siddle to Clarke at slip and the same bowler then had James Anderson caught by Hughes at mid-wicket for a duck.
Openers Shane Watson and Chris Rogers then survived seven overs before lunch without any major alarms to reach the interval on 28 for no wicket.
The pair also batted confidently through the first hour of the afternoon session as England struggled for inspiration under blue skies.
Watson peppered the boundary with eight fours and he looked certain to reach his half-century before a lapse in concentration allowed Broad to trap him lbw for 46 with a full-length delivery, ending a stand of 84.
The opener referred the umpire's decision to give him out but replays showed the ball would have hit the stumps and he left the field shaking his head after once again failing to turn a good start into a really significant contribution for his team.
Ed Cowan, on a king pair, got off the mark with a cut for four off Steven Finn and the 35-year-old Rogers reached his first test half-century, off 104 balls and including eight fours.
Cowan was out for 14 three balls before tea, rashly trying to drive a wide ball from part-time spinner Joe Root and edging a simple catch to Jonathan Trott at slip to leave Australia on 111 for two.
Rogers, on 52, tried to work Anderson through the leg side and chipped a simple catch to Bell at mid-wicket.
Clarke and Smith dug in as the tension mounted and survived a couple of huge appeals before the skipper, on 23, feathered Broad through to wicketkeeper Matt Prior.
The umpire checked to see whether the ball had carried and Clarke then referred the decision, only for it to be confirmed by the television official.
With the crowd still celebrating, Smith was trapped lbw for 17 by Swann and three runs later the England off-spinner removed Hughes for a duck in identical fashion.
Agar, promoted up the order after his superb 98 at number 11 in the first innings, came to the crease and the 19-year-old again showed great composure on his test debut.
He and Haddin batted through the remaining overs with few alarms and Australia will still feel they have a realistic chance of winning this remarkable test.