But despite being outclassed in nearly every department, England were utterly superior at scrum time and referee Nigel Owens had little choice but to award them two second-half penalty tries.
Yet another piece of magic from Cooper and the boot of the impressive James O'Connor kept the visitors from gaining their first victory on Australian soil since the 2003 World Cup final.
Australia were utterly dominant in the opening quarter of the half but O'Connor - hat-trick star of the Australian Barbarians' 28-28 midweek draw - missed an early penalty and Chris Ashton executed a try-saving tackle on Drew Mitchell kept the game scoreless after the opening quarter.
But the dam was breached on 17 minutes when Mitchell led a counter attack before the ball was superbly recycled by O'Connor for captain Rocky Elsom to gleefully touch down.
Full-back O'Connor, just 19, added the conversion but for all their territory and possession Australia's Achilles Heel was at the scrum with their first-choice front three missing.
Yet from their first kicking opportunity after a Wallabies' offence at the scrum, Toby Flood's penalty was just short.
For all their woes up front, Australia on the attack were irresistible and scrum-half Luke Burgess's superb break and inside ball to Cooper allowed the fly-half to nonchalantly touch down one handed. O'Connor's conversion put the hosts 14-0 ahead.
England's one real period of pressure came towards the end of the half but there was no way through the Green & Gold defence.
Four minutes into the second offence, another Wallabies' scrum offence finally allowed England to get on the scoreboard via a Toby Flood penalty and then a brilliant break led by Ben Foden and Chris Ashton led to the visitors being held up just short of the line.
Inevitably England opted for a series of scrums which resulted in Owens awarding his first penalty try but no sooner than they had got back into than Australia immediately responded.
Digby Iaone made ground before Cooper arrived on his shoulder to finish things off. Cooper's conversion made it 21-10.
England looked far more dangerous with ball in hand but their ultimate weapon was the scrum and it was little surprise with ten minutes remaining when Owens awarded a second penalty try.
With just four points in it and Australia's old foe Jonny Wilkinson on the field, the stage was set for a grandstand finish but ultimately it was the home side who held their nerve as O'Connor and Cooper kicked late penalties to take the Wallabies out of sight.