Nathan Lyon's magnificent seven-for stymied Bangladesh before gritty fifties from David Warner and Peter Handscomb contributed to a strong second day for Australia in the searing heat of Chittagong.
Bangladesh resumed on 253-6 but could only add a further 52 runs as Lyon's fine form continued, the off-spinner claiming the key wicket of captain Mushfiqur Rahim (68) as he returned figures of 7-94.
Australia suffered an early setback with Matt Renshaw departing off the ninth ball of their innings for just four and skipper Steve Smith's exit after a commendable 81-ball fifty brought further concern.
However, Warner (88) and Handscomb (69) put on a 127-run partnership for the third wicket that proved impenetrable for Bangladesh's bowling attack, despite the batsmen visibly fatiguing in the final session.
Warner showed incredible patience on a day that uncharacteristically saw him hit just four boundaries, but his efforts helped ensure Australia reached stumps at 225-2, trailing Bangladesh by 80 runs.
Lyon had already taken five wickets for the third Test in succession but he only needed two balls to add to his tally, immediately finding turn that prompted Mushfiqur to play on.
With their captain gone, Bangladesh offered little resistance and Lyon had them all out for 305 when Taijul Islam (9) edged to Smith at slip.
Renshaw had barely got going when wicketkeeper Mushfiqur sprang to his right to snaffle his glancing shot one-handed off Mustafizur Rahman (1-45).
The experienced duo of Warner and Smith laid a solid foundation, though, the latter finding the boundary eight times in a 93-run union that was finally broken before tea. Left-arm spinner Taijul (1-50) was the man responsible, finding the gap between Smith's bat and pad.
Australia were forced to play defensively and Warner's 25th Test fifty came off 98 balls - his fourth-slowest in the longest format - before Mominul Haque dropped him at short leg on 52.
The tourists ground out the runs after that reprieve and typically explosive opener Warner had to endure what will surely have been a painful 74-ball wait between boundaries.
Handscomb survived an lbw review after it was deemed he got an inside edge before completing his half-century, a relief after disappointing showings in his last two Tests.
The batsmen continued to pick Bangladesh off and, after their stand passed 100 runs, Mushfiqur missed a chance to stump Warner.
The 30-year-old ventured down the track only to miss Mehedi Hasan's low delivery, but the wicketkeeper could not react quickly enough and the ball bounced off his pad to let the Australian off the hook.
Warner managed to dig in until stumps, with Bangladesh in need of improvement if they are to stop Australia fighting on during day three.