Cook, who was unbeaten on a career-high 235, and Jonathan Trott, who was 135 not out, had built a record partnership of 329 when captain Andrew Strauss declared England's second innings closed at 571-1 before tea.
Australia and England drew the first Ashes test on Monday but the tourists will claim a moral victory after Alastair Cook dominated the last two days with an unbeaten double century to rescue a match that had looked lost.
Cook, who was unbeaten on a career-high 235, and Jonathan Trott, who was 135 not out, had built a record partnership of 329 when captain Andrew Strauss declared England's second innings closed at 571-1 before tea.
Australia, who were 296 runs behind, lost Simon Katich before the break but skipper Ricky Ponting, who hit 51 for his 56th test half century, and opener Shane Watson (41) ushered the hosts safely through the remaining overs to finish on 107 for one.
Cook and Trott completely dominated the first two sessions of the day in front of a sparse fifth day crowd at the Gabba, where England supporters were more numerous and much louder.
Their disciplined partnership bettered the 307 Mike Hussey and Brad Haddin achieved to put Australia firmly in charge on Saturday and the 188 Cook and Strauss, who made 110, managed on Sunday.
It was the highest stand by Englishmen in Australia, beating the 323 achieved by Jack Hobbs and Wilfred Rhodes on the 1911-12 Ashes tour.
The "Barmy Army" were at their noisiest just before lunch when the 25-year-old Cook clipped the ball behind square to become the fourth England player to score an Ashes double century in Australia.
They were back on their feet minutes later when Trott pushed a shot through leg side and scuttled for three runs, leaping into the air in jubilation as he raced to safety and his fourth test century.
Australia's bowlers continued to struggle in the baking sunshine with the one cast-iron chance they had of separating the England batsman before lunch spilled in the slips.
All-rounder Watson, the pick of Australia's bowlers, carved out the opportunity when Trott was on 75 but Michael Clarke could not hold on to a simple catch.
Ponting thought he had caught Cook caught out on 209 but the umpires conferred and decided the ball had been grounded, and he missed a difficult slip catch when the Englishman was on 222.
Fast bowler Mitchell Johnson's troubles continued as he ended the test wicketless at the cost of 170 runs.
One wretched delivery to Trott hurtled some three metres wide of the wicket in echoes of England bowler Steve Harmison's first delivery here four years ago that set the tone for England's 5-0 series defeat.
Simon Katich lasted just 23 minutes before England succeeded where Australia's bowlers had failed, the opener caught by Strauss in the slips off the bowling of Stuart Broad for four runs.
The second of the five tests begins in Adelaide on Friday.
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