The visitors were 111-2 at tea in their first innings, still trailing by 194. David Warner on 42 and Peter Handscomb on five were at the crease.
Opener Matt Renshaw departed early, dismissed by fast bowler Mustafizur Rahman before the lunch break.
The left-hander tried to tickle a sliding delivery down the leg side but wicketkeeper Mushfiqur Rahim took a stunning catch.
Warner was then involved in a 93-run second-wicket partnership with Smith as the pair steadied the innings.
But Smith, who registered his 21st Test half-century, was done in an arm ball from the left-arm spinner which sneaked through the bat-pad gap to hit middle stump.
Earlier Australian off-spinner Nathan Lyon claimed seven wickets to help bowl out the hosts for 305 in the first session.
Bangladesh lead the series 1-0 after registering their first-ever Test win against Australia last week
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