England managed to set their hosts only 231 to win, after another batting collapse on day three here, and on the fourth morning Australia raced from 30 for none to a lunchtime 143 for one.
Rogers (81 not out) was their cornerstone as the prospect of a whitewash, in Sydney next week, loomed larger as the likely end product of England's miserable tour.
They missed two chances on a fresh morning to break Rogers' opening stand of 64 with David Warner, on 39 and then 52.
Rogers had 19 when he edged a good delivery from Stuart Broad, but the ball flew between Jonny Bairstow and first slip Alastair Cook - who did get fingertips but no more to it as four runs were instead conceded.
It was a moot point whether wicketkeeper Bairstow could have tried to make the catch his own.
Two overs later, after Ben Stokes replaced Broad at the Members End, Cook had only himself to blame when he dropped a regulation slip catch to reprieve Warner on 22.
England did make the breakthrough soon afterwards when Warner tried to upper-cut Stokes but got only a thin edge through to Bairstow.
Rogers stood firm, though, for a valuable 73-ball fifty.
He was not always convincing, for example when he went to 49 with his sixth four via an involuntary inside edge off Stokes which snaked barely an inch wide of off stump and then past a diving Bairstow.
But for Australia, Rogers was getting the job done - and when Shane Watson joined in with successive fours driven off Tim Bresnan, in another half-century stand, it was hard to escape the conclusion of a surge to home victory achieved without significant snags.
Cook's deployment of his bowlers was hard to fathom, front-line spinner Monty Panesar kept out of the attack while part-timer Joe Root was given two spells - and Broad unbowled, until 20 minutes till lunch, after two initial overs in which he had Rogers dropped.
Whoever bowled, though, England were simply not threatening wickets.