Australia, however, were well-placed at 184 for two at tea on the first day.
Future captain Steven Smith was 29 not out and current skipper Michael Clarke, in his last match before his international retirement, 14 not out.
Rogers, also playing his last Test before calling time on his Australia career, and fellow left-handed opener Warner put on 110 after Cook won the toss on a green-tinged pitch and under overcast skies.
England went on to win that match by an innings and 78 runs, a victory that saw them regain the Ashes and take an unbeatable 3-1 lead in the five-match series.
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But Rogers, one of the few Australia batsmen who had been willing to 'knuckle down' this series, and Warner were far more watchful on Thursday.
Meanwhile Stuart Broad, who took a sensational eight for 15 in Australia's first innings at Trent Bridge, was fractionally too wide against batsmen determined not to 'chase' the ball after their Nottingham nightmare.
Australia's first boundary did not arrive until the 15th over when the normally aggressive Warner -- who had faced 45 balls without a four -- pulled Ben Stokes through square leg.
An increasingly assured Warner, gaining from time in the middle, went to a 76-ball fifty when he cover-drove fast bowler Steven Finn for his seventh four.
At lunch, Australia were 82 without loss, with Rogers 43 not out and Warner 53 not out.
England eventually separated the pair when Rogers (43) at last pushed away from his body against fast bowler Mark Wood and first slip Cook held the outside edge at the second attempt.
The opener swished his bat in annoyance, having added 51 for the with Smith for the second wicket to take Australia to 161 for two.
New batsman Clarke walked out to a standing ovation from the crowd as England's players formed a guard of honour in the middle for the Australia captain.