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The tourists began play on 180-4, hoping to get close to Australia's first-innings total of 385, but lost 6-71 in 20 overs as they again failed to pass 300.
David Warner's second century of the series then helped the hosts establish a 369-run lead at the close as they left the field on 235-3 in the knowledge that the game was likely beyond their struggling opponents.
Brad Haddin claimed two more scalps as Matt Prior (9) and Tim Bresnan (21) edged off Peter Siddle and Johnson respectively, either side of the dismissal of Stuart Broad, who was sent to hospital for an X-ray on his foot after being struck on his boot in the process of getting out lbw to Australia's left-arm quick.
With Broad unfit to bowl, England's stretched attack was put to the sword as the new-ball partnership of Bresnan and Anderson were dispatched to all parts by Warner, and Chris Rogers, who made 54.
Prior missed two stumping chances against Warner off Swann, while Cook dropped Rogers in the slips and the opening pair punished the tourists by putting on 157 for the first wicket.
Michael Carberry ended England's torture by catching Rogers well off Bresnan, and Warner, who took his individual run tally for the series up to 457 with a brilliant 140-ball knock of 112, followed soon after.
Swann had him snared on the boundary by all-rounder Stokes, who produced an excellent ball that nipped back to bowl Aussie captain Michael Clarke (23) in his 100th Test just before the close.
Shane Watson (29*) and Steve Smith (5*) will resume at the crease tomorrow morning, with Australia expected to bat for another two hours and then force England to bat for five sessions to save the match.