Australia were 424 for three at lunch on the second day, with Smith 168 not out after Rogers had fallen for 173.
Adam Voges was 24 not out.
England, led by fast-medium bowler Stuart Broad, restricted Australia to a session run-rate of under three an over for the loss of both Rogers and captain Michael Clarke.
But the bulk of the damage had been done on a first day where Australia posted 337 for one.
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Then, as often happens after a lengthy wait, the new batsman fell cheaply with Clarke out for seven.
Veteran left-handed opener Rogers had already bettered his previous highest Test score of 119, made against England at Sydney last year, after batting through the whole of yesterday's first day for 158 not out.
The 37-year-old, who spent several seasons at Lord's with county side Middlesex, had a worrying start on Friday.
The very first ball of the day's play saw him struck flush on the side of the helmet as he turned his head away from a James Anderson delivery which didn't bounce as much as he expected.
Two balls later, however, he square-drove Anderson for four.
It was Broad, Anderson's new ball partner, eventually pierced Rogers's defence with a nip-back ball that took the inside edge and pad before crashing into the stumps to leave Australia 362 for two.
Rogers batted for over six-and-a-half hours, faced 300 balls and struck 28 fours.
His stand with Smith topped the previous Australia record Test partnership at Lord's of 260 shared by openers Mark Taylor and Michael Slater in 1993.