Cameron Green (43 not out) and Alex Carey (19 not out) dug in to keep Australia afloat, but the hosts face a huge task to first erase the deficit and then build a competitive total to defend.
Bangladesh could even secure an innings victory if they take six more Australia wickets quickly, which would secure their first Test victory on Australian soil and arguably their biggest win in Test cricket.
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Resuming at 351-6, Mehidy Hasan Miraz calmly shepherded Bangladesh's tail to another 75 runs, taking the visitors to 426 all out and extending their first-innings lead beyond 200, before returning to make the crucial breakthrough with the ball by removing Steve Smith late in the day.
Mehidy shared partnerships of 46 with Hasan Mahmud, 18 with Taijul Islam and a useful 46 with Taskin Ahmed before eventually falling for 65 from 154 balls.
Australia were their own worst enemy at times with three dropped catches, including an inexplicable miss by Smith in the deep, taking their tally for the match to five.
Josh Hazlewood claimed all four wickets to fall on Saturday and finished with 6-89 from 28 overs.
His dismissal of Mehidy was also his 300th Test wicket, making him the fourth member of Australia's famed "Big Four" bowling attack to reach the milestone and the ninth Australian overall.
Australia lost both openers cheaply as Bangladesh tightened their grip on the match at Marrara Stadium before tea.
Jake Weatherald (0) and Travis Head (17) both chopped on attempting to cut paceman Hasan Mahmud as Australia slumped to 37-2.
It took Mahmud's match tally to eight wickets after his first-innings return of 6-55 helped dismiss Australia for 198 and set the tone for the tourists' dominance over the following two days.
Marnus Labuschagne, also searching for runs after a lean spell, looked busy at the crease until he missed a straight ball from spinner Taijul Islam (1-28) and was bowled for 31 from 59 deliveries.
Australia's hopes of salvaging the match rested on Smith's shoulders and the veteran looked composed as he characteristically accumulated his runs with minimal risk.
The match swung firmly in Bangladesh's favour, though, when Mehidy (1-20) snaffled a sharp caught-and-bowled chance to dismiss Smith for 44, leaving Australia 122-4 and still 106 runs in arrears.
The largest first-innings deficit Australia have overcome to win a home Test is 206 runs, achieved against Pakistan at the Sydney Cricket Ground in 2010.
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