Nathan Lyon claimed 7-94 as Bangladesh were all out for 305 runs earlier on the second day.
David Warner remained not out on 88 as Australia finished the second day at 225 for 2 in reply to Bangladesh's first-innings total of 305 in the second and final test on Tuesday.
Warner featured in two significant partnerships to lead the visitors' innings as they press for a pride-salvaging victory in Chittagong after losing the first test in Dhaka.
Warner, in contrast with his explosive batting style, hit only four boundaries in his patient knock as he kept Australia on course for a handy first-innings lead. Peter Handscomb, Warner's partner in their unbroken 127-run stand, braved Bangladesh's relentless spin assault and dehydration to remain not out on 69. His fourth test fifty included five boundaries.
Earlier, Nathan Lyon claimed 7-94, his second-best figure, as Bangladesh added 52 runs to their overnight score of 253-6 before collapsing.
The off-spinner, who had claimed five wickets on Monday, struck with his second delivery of the day, dismissing Mushfiqur Rahim for 68. Ashton Agar dismissed the other overnight batsman Nasir Hossain for 45 with Matthew Wade, whose place in the playing XI was in doubt going into the test, taking a fine catch.
Warner then ran out Mehidy Hasan before Lyon dismissed Taijul Islam to claim his seventh wicket.
When Australia returned to bat, Mushfiqur took a spectacular diving catch to dismiss Matt Renshaw for four but Warner and Steve Smith raised 93 runs for the second wicket to overcome the loss.
Warner defended resolutely, letting Smith dominate their partnership, before the Australia skipper was bowled through the gate for 58 by spinner Taijul Islam. Smith hit eight boundaries in his fluent knock. New man in Handscomb proved an able ally as Warner frustrated Bangladesh with his compact defence, scoring largely through singles and twos, and he was upset with himself for not carrying on.
The heat and humidity took a toll on Handscomb who looked dehydrated and needed medical attention towards the end before resuming batting. Rahim wasted a stumping opportunity when Warner, then on 73, charged out to Mehidy Hasan and missed the ball which kept low and hit the stumper's pad.
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 "David Warner Australia vs Bangladesh")
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