Australia were 104 for one at lunch on the first day, with David Warner the only batsman dismissed, as they looked to level the five-match series at 1-1 after their 169-run defeat in the first Test in Cardiff last week.
Rogers was 43 not out, on a ground he knows well from his time with Lord's-based county side Middlesex, and Steven Smith 16 not out.
Australia captain Michael Clarke won the toss and elected to bat first on what looked another typically good pitch at the 'home of cricket'.
The first over saw Rogers fortunate to survive when he edged an intended drive off James Anderson just over the top of the slip cordon for four.
Warner is known as the more aggressive of Australia's two left-handed openers.
But it took him 25 balls to score his first four on Thursday, pulling first-change Mark Wood.
Rogers did not look secure and repeatedly played and missed
By contrast, Warner started to find his touch with five fours, all off fast bowler Wood, in 13 balls.
England captain Alastair Cook brought on Moeen Ali after drinks.
Ali, however, had his revenge five balls later as he dismissed Warner for the second time in as many innings.
Warner, going down the pitch, saw a lofted drive well-caught by Anderson on the run at deep mid-off and was out for a near run-a-ball 38 that included seven boundaries.
Australia were now 78 for one off exactly 15 overs.
England fielded the same side that won in Cardiff after all-rounder Ali overcame a slight side strain.
However, Ashes-holders Australia -- looking for their first Test series win in Britain in 14 years -- made two changes.
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