Cook (244no), with a best of just 37 in his previous 10 innings, resumed on day three with his 32nd Test hundred already in the book - and what followed was remarkable, especially during a ninth-wicket stand of exactly 100 with Stuart Broad (56), as he made up for the fragility of several team-mates.
England’s all-time record run-scorer beat Wally Hammond’s previous ground-record 200 for a compatriot, and then surpassed Viv Richards’ best of 209 from any overseas batsman here.
For England, the bottom line of 491-9 and a first-innings advantage of 164 were riches indeed after captain Joe Root (61) and others had briefly invited Australia back into the contest.
Cook also moved up to sixth in the global list of top Test run-scorers just before stumps, above the great Brian Lara, having already passed his double-hundred by straight-driving his 360th ball for his 23rd four off Jackson Bird after almost nine-and-a-half hours at the crease.
How the tourists needed his sustained return to form, too, as they try to avoid a series whitewash, after losing the Ashes in Perth last week.
Root was rightly furious with himself when, as has become his unwelcome trademark, he failed to convert his 50 into a century - guilty of both faulty shot selection and execution, mis-pulling Pat Cummins to deep square-leg.
He stormed off, kicking the pitch on his way and ditching his batting gloves as soon as he made it past the boundary marker.
Root’s misadventure was far from England’s last, Dawid Malan failing to invoke DRS over his lbw dismissal despite the fact that - as with James Vince the previous day - Hotspot indicated an inside-edge on a delivery from Josh Hazlewood.
Jonny Bairstow and Moeen Ali then both got out to Nathan Lyon in the middle session.
Moeen was easily the most culpable, but Bairstow’s departure - caught behind making room to cut the off-spinner - was avoidable, too.
Moeen’s manic cameo spoke of a cricketer who has got himself in a proper tangle on this tour, before he hit a long-hop to cover to go for 20 from just 14 balls.
England needed a much more sensible hand from Chris Woakes, to stop a costly collapse taking hold, and he was perfectly on-messsage.
Cook was dropped by Steve Smith for a second time - having given a sharp slip chance on 66 on day two - and this time he profited when the home captain could not hold on low down at square-leg off Cummins.
Woakes was bounced out by Cummins straight after tea to end a crucial partnership of 59 - but one which merely prefaced the improbable Broad-Cook show.
After debutant Tom Curran edged Hazlewood behind, it seemed Cook would be up against it to complete his double as the home pace attack instantly targeted Broad with the short ball.
There were several scrapes, of course, but skill and judgement too until Broad fell in mild controversy to Usman Khawaja’s diving and juggling outfield catch - having done much, with his eight fours and a six from 63 balls, to give England an outstanding chance of reducing the series arrears to 3-1.