Jake Weatherald (0) and Travis Head (17) both chopped on to paceman Hasan Mahmud to leave the hosts in danger of a shock defeat at Marrara Stadium.
Steve Smith (8no) and Marnus Labuschagne (21no), who both had nervous moments, were at the crease at tea, with Australia 2-51 and still trailing by 177 runs.
Australia will have to rewrite the Test history books on home soil if they want to beat Bangladesh after the visitors took a first-innings lead of 228.
Bangladesh allrounder Mehidy Hasan Miraz (65) and the tail took the total to 426 all out after lunch on Saturday.
Australia paceman Josh Hazlewood was the shining light with the ball, taking 6-89. The wicket of Mehidy was his 300th in Test cricket as he became the ninth Australian to reach the milestone.
The Bangladeshis finished with their sizeable lead after Australia collapsed to be all out for 198 on day one.
The largest first-innings deficit Australia have come back from to win a Test on home soil was 206, when they trailed Pakistan at the SCG in 2010.
On that occasion Michael Hussey made an unbeaten 134 in the second innings and Australia bowled Pakistan out for 176 to win by 36 runs.
Australia's woes were summed up before lunch on Saturday when No.10 batter Taskin Ahmed hooked Pat Cummins for four with disdain. Next ball Steve Smith dropped a sitter at deep backward square leg.
Head then spilled a skied pull shot off Mehidy, a difficult chance, to make it five dropped catches for the innings by Australia.
The visitors were in no rush, clearly determined to keep the Australian bowlers out in the heat for as long as possible.
Mehidy successfully reviewed an lbw decision against him in Hazlewood's first over.
Fast bowler Hasan Mahmud (14) wasn't reprieved when he sent upstairs a Hazlewood delivery that was smashing into middle stump. The 26-year-old had defied the bowlers for 92 deliveries after his brilliant 6-55 in the first innings.
Spinner Taijul Islam (17) took a more aggressive approach, flipping Hazlewood over the square leg boundary before slashing the same bowler to gully, in a handy cameo.
Hazlewood vindicated his selection ahead of the unlucky Scott Boland and was the pick of the bowlers.
Bangladesh's first-innings lead was set up by a maiden ton for Tanzid Hasan (101) and an aggressive 84 by captain Najmul Hossain Shanto.
Australian legend Matthew Hayden told Channel Seven that a win was beyond the hosts.
"At 220 (behind) ... you've got a completely different mindset as a batting unit," Hayden said at lunch.
"You are just looking to survive. That is it. You're not looking to win. It's a survival. And that's a real struggle for Australian cricket. We do not like to be in that position."
Australia allrounder Beau Webster, speaking at the end of day two, said the path to victory was not insurmountable.
"Hopefully the wicket stays true and nice and we can put some pressure back on their bowlers," Webster said.
"A day and half of really good batting at a half-decent clip gets us right back in this contest.
"Batting last in a Test match on a hopefully worn wicket could be tricky (for Bangladesh) if it is a small chase.''
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