The tourists, replying to England’s 430 all out, also lost David Warner (17), Steve Smith (33), captain Michael Clarke (38) and Adam Voges (31).
Smith and Clarke were coaxed into errors by the spin of Moeen Ali who hit an entertaining 77 as the home team kept the Australian fielders busy for 75 minutes in the morning, adding 87 to their overnight total for the loss of the last three wickets.
The 37-year-old Rogers has been superbly consistent in a run of 50-plus scores that began against India in Brisbane in December but he will kick himself as a century appeared there for the taking.
He had progressed largely untroubled towards a fourth Ashes hundred when he tried to cut a ball from paceman Mark Wood that was too close to him and nicked it behind to wicketkeeper Jos Buttler, trudging forlornly off the field at Sophia Gardens.
A deserved 100 eluded him but he still joined a select band with a seventh successive knock of 50-plus, matching West Indians Everton Weekes and Shivnarine Chanderpaul, Zimbabwean Andy Flower and Sri Lankan Kumar Sangakkara.
Rogers and Warner had begun well, sharing a 52-run opening stand before the latter was well caught at first slip by England captain Alastair Cook off James Anderson.
Moments later Cook, much to team mate Joe Root’s mirth, was rolling around in agony after being struck a painful blow in the groin area that forced him off for treatment.
Smith, the world’s number one test batsman, clubbed Moeen to the boundary three times in one over but the spinner had the last laugh by inducing a leading edge that Cook caught at short mid-on, ending a stand of 77 with Rogers.
Clarke also looked set for a big score before dancing down the track to Moeen and offering a sharp return catch.
Australia would have been fairly happy reaching the close of play four wickets down but a late lapse in concentration from Voges, who played a loose drive off Ben Stokes and was caught at short cover, left England feeling they could establish a decent lead.
Earlier, Mitchell Starc shook off a sore ankle to finish with five for 114. Fellow paceman Mitchell Johnson, the scourge of England in their Ashes demolition in Australia last year, conceded 111 runs and went wicketless.
(Editing by Tony Jimenez)