At tea, Australia were 145 for two in reply to England's 430, a deficit of 285 runs.
Veteran opener Rogers was 74 not out and Australia captain Michael Clarke 11 not out.
Rogers missed Australia's 2-0 series win in the Caribbean with Caribbean and the 37-year-old left-hander has said the Ashes will be his final Test campaign before retirement.
Shortly after lunch, England dismissed Australia dangerman David Warner for 17 when the left-handed opener got a thick edge off James Anderson and England captain Alastair Cook held a sharp catch at first slip.
Smith, now ranked as the world's number one Test batman, got after Moeen Ali's off-breaks with three fours in four balls, the first a classic drive from a batsman often described as "unorthodox".
England, lacking the sheer pace of Australia's attack, found wickets hard to come by in near-ideal batting conditions at a sun-drenched Sophia Gardens.
But Ali succeeded where England's paceman had failed by dismissed the in-form Smith for 33.
The world's number one ranked Test batsman got himself in a terrible tangle to avoid being stumped as Ali responded to his charge by sending the ball legside.
