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The former England captain strode off the Melbourne Cricket Ground on 244 not out, the highest ever score by a touring batsman at the venue, having pushed his side to a first innings lead of 164 runs.
Stuart Broad provided gallant support with a swashbuckling 56, sharing a 100-run ninth-wicket partnership that crushed the spirits of a demoralised attack and trampled on Australia's hopes of whitewashing the series.
The urn may be gone with England an irretrievable 3-0 down in the five-match series, but Joe Root's team will savour the finest day of the campaign and are well-placed to grab a consolation win.
Cook smashed a straight drive past bowler Jackson Bird for his 23rd four to raise his fifth double-century in 360 balls, triggering a standing ovation from the crowd of 61,839.
It was not a chanceless innings, however, with Australia captain Steve Smith dropping him on 66 on Wednesday and again on 153.
The pitch was a batsman's paradise and the Australian attack shorn of injured spearhead Mitchell Starc, but England's most prolific run-scorer may remember few more gratifying innings in his 151 tests.
"Probably one of the more emotional (ones)... from where I'd been on this tour," Cook, who was woefully out of form prior to Melbourne, told reporters of his 409-ball knock.
"It meant a lot last night and then today I was quite proud that I managed to back it up after all the emotion came out yesterday. To get a real big one for the team was really important."
Cook, who resumed on 104 not out, capped a memorable day with a sumptuous cover drive for four off all-rounder Mitchell Marsh, lifting him past Brian Lara to sixth on the all-time test runs list.
Broad was brave in support, wearing a short-pitched battering before paying Australia's pacemen back in an counter-attacking innings of eight fours and a six.
He bookended his fifty with two fours off Pat Cummins before holing out with a miscued pull that was caught in the deep by a diving Usman Khawaja, with replays suggesting the fielder may have spilled the ball onto the turf as he rolled over.
England resumed on 360-6 after tea, before Chris Woakes quickly fell for 26 after gloving a catch behind from a clumsy attempt at a pull shot and debutant Tom Curran added just four runs before edging Josh Hazlewood to wicketkeeper Tim Paine.
Broad came to the wicket and was hit on the shoulder by a bouncer from paceman Hazlewood but the tail-ender weathered the barrage to slog England past 400.
England's batting had been brittle throughout the series, but for once the wickets fell slowly rather than in clumps.
Cook and Root had built a 138-run partnership before the England captain fell for 61 in the morning, mistiming a pull shot straight to Nathan Lyon at deep square leg.
Dawid Malan squandered his wicket for 14 when he failed to review an lbw decision despite edging the ball from Hazlewood.
It was the second England wicket to fall lbw with a nick in the innings, with number three James Vince failing to review his dismissal on the second day.
Off-spinner Lyon struck twice after lunch to reduce England to 307-6, with Jonny Bairstow (22) and Moeen Ali (20) both falling to poor shot selections.
Having dismissed Moeen for the sixth time in the series, Lyon gave the all-rounder a generous verbal spray as he trudged off after driving in the air to Shaun Marsh at short cover.
Cook and Woakes then steadied England with a vital 59-run partnership that pushed them past Australia's first innings 327 before Broad helped extend the lead out past three figures.
Australia coach Darren Lehmann conceded the hosts' whitewash bid was all but over.
"We'll have to get 450 in a day. How do you want us to do that? Slog?" he said.
"From our point of view we'll just be hopefully batting the day and reassessing what happens after tomorrow."