Australia captain Michael Clark is likely to bat until around lunchtime on Sunday and set England an unlikely 500 over five sessions.
While Australia bowled well, England's batsmen were often the cause of their own downfall. Several of them displayed technical deficiencies that will worry new coach Trevor Bayliss, the Australian briefed to try to regain the Ashes for England.
Victory for the tourists at Lord's would level the series 1-1, with three tests to play.
Rogers scored 173 in the first innings, with Steve Smith scoring 215 before Australia blew away England's top order on Friday evening, leaving the hosts on 85-4 at the close of day 2.
On Saturday, England put on a slightly better show, but only captain Alastair Cook and allrounder Ben Stokes were able to stick around. Cook was dismissed four runs short of his century and Stokes scored 87 as several England batsman displayed frailties against the aggressive and disciplined Australian pace attack.
Stokes said England tried to play positively on Saturday but will need to bat for about 150 overs in the second innings. "We're under the pump, tomorrow's a big day for us," he said. "It's pretty much over to the batters for the next two days."
Cook and Stokes shared a partnership of 145 but Australia's bowlers remained patient and grabbed the crucial wickets of Stokes just before lunch and then Cook shortly before tea - just short of what would have been his 50th first-class century.
Mitchell Johnson was the pick of the Australian bowlers, with 3-53 from 20.1 overs.
Josh Hazlewood also took three wickets.
"We'll obviously want a few more runs (before declaring)," Hazlewood told reporters. "We're getting them pretty quickly at the moment, so hopefully that keeps up tomorrow. I think we'll need a little bit of time to bowl these guys out on a pretty flat wicket.
"Batting first was key and we did the job with the ball yesterday and today. So we're in a pretty good position."
Cook was batting impressively and looking increasingly assured when he attempted to drive a wide delivery from Mitchell Marsh. The England captain was deceived by the pace and got a thick inside edge onto his stumps with the score on 266.
He batted for 233 balls and nine minutes short of six hours, hitting 13 fours.
Two wickets fell in the afternoon session. Wicketkeeper Jos Buttler never looked comfortable against the pace of Johnson and Mitchell Starc.
Australia wicketkeeper Peter Nevill was convinced he had held a diving catch with the score on 186-5, but Buttler did not budge and replays showed the ball touching the ground as Nevill caught it.
Buttler did not last long.
With the score on 210, he played and appeared to miss a delivery from spinner Nathan Lyon. Umpire Kumar Dharmasena did not initially give Buttler out and replays appeared to show there had been no contact, but Buttler walked voluntarily, effectively giving himself out.
Cook and Moeen Ali put on 56, with Ali playing some belligerent shots that included a straight six off Lyon, before the England opener was dismissed by Marsh. Ali scored 39 and his departure enabled Australia to wrap up the English tail.
