The Australians were dismissed for 138 off 58 overs at the break and after leading by 215 runs on first innings pushed their lead past historical proportions at the ground.
England will have to create history to level the series after losing the first Brisbane Test by 10 wickets last week.
England's all-time leading wicket-taker James Anderson took five for 43 -- his first five-wicket haul in Australia -- while fellow paceman Chris Woakes captured four for 36.
Usman Khawaja and Mitchell Starc were joint top scorers with just 20 each.
England kept the Australians under pressure after they resumed at 53 for four on the fourth day. Nightwatchman Nathan Lyon hit a couple of fours and was struck on the helmet grille by Stuart Broad before he holed out to mid-off off Anderson for 14 for the first wicket to fall on Tuesday.
Peter Handscomb, the other overnight batsman, struggled before he aimed to steer through the slips cordon only to be caught by Dawid Malan for 12 in 76 minutes to give Anderson his fourth wicket.
Australia took their overall lead past 300 with four overthrows after Mark Stoneman's attempt to run out Shaun Marsh missed the stumps at the non-striker's end and ran to the boundary.
Tim Paine, who scored a half-century in the first innings, was dismissed for 11 top-edging a pull shot off Woakes and Craig Overton took a splendid sprawling catch running in from fine leg.
Mitchell Starc launched a six over mid-wicket off-spinner Moeen Ali before Marsh was bowled by Woakes for 19, ending a 32-run partnership with Starc.
Anderson made a valiant effort to catch Starc on 18, haring down the pitch off his own bowling but he could not hold a difficult two-handed chance as he slid into the stumps, tearing his trousers.
But Anderson got five wickets in Australia for the first time when Starc holed out to Moeen for 20.
Josh Hazlewood was the last wicket to fall, caught in the gully off Overton for three, leaving Pat Cummins not out 11.
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