The Australians went to lunch at 92 for two after Usman Khawaja fell for one when he was adjudged to have been caught behind off Graeme Swann.
The batsman reviewed the decision but the third umpire sided with his on-field colleague despite replays showing no obvious edge in the latest decision review system dispute to afflict the series and Australia.
Shane Watson earlier got away with several loose shots through the slips, over gully and just short of point but was finally caught at slip by Alastair Cook off paceman Tim Bresnan for 19.
Rogers, obeying the message from on high to knuckle down and build an innings after repeated Australian carelessness this series, calmly compiled his highest test score to silence doubters of his worth to the side.
The 35-year-old, selected because of his knowledge of English conditions having been a mainstay of the county scene, is playing in his fourth test only for Australia. Captain Michael Clarke was five not out.
England, who lead the five-match series 2-0 and can retain the Ashes at the revamped Manchester venue with a win or a draw, extracted decent swing under slightly overcast skies and to the accompaniment of a lone trumpeter in the packed crowd.
Stuart Broad had one lbw shout against Watson, out so often that way, but England decided not to review the umpire's not out call and later made the breakthroughs just as Australia looked to be taking the initiative.
Australia had lost the toss at Trent Bridge and Lord's, when England batted first both times on their way to victory, and the relief on Clarke's face was visible as the coin landed his way.
The tourists, who risk losing a seventh test in a row and a third straight Ashes series, brought in David Warner at number six after the aggressive left-hander returned from his banishment to the A squad for punching England's Joe Root in a bar in June.
He replaced the ineffective Phil Hughes while off-spinner Nathan Lyon came in for left-armer Ashton Agar and paceman Mitchell Starc replaced the injured James Pattinson.
England were unchanged after Kevin Pietersen passed a fitness test on a calf strain.
(Editing by John Mehaffey)
