Bangladesh had reduced Australia to 145-6 by stumps on the second day after scoring 427 on Monday. The visitors need another 83 to avoid the follow-on.
Adam Gilchrist (51) and Brett Lee (13) were the not out batsmen as Bangladesh's spinners held centre-stage after their Australian counterpart Stuart MacGill had produced his best performance in Tests.
Australia were fresh from their memorable 3-0 Test series win in South Africa but struggled in totally different conditions.
The ball kept low when they came in to bat and Bangladesh's left-arm spin-pair of Mohammad Rafique and Enamul Haque struck telling blows after early breakthroughs by the new-ball bowlers.
Rafique bowled Damien Martyn (four) with a ball that skidded through and then castled opener Michael Hussey (23) between bat and pad.
Haque bowled Michael Clarke (19) and had Shane Warne (six) caught behind by Khaled Mashud as the visiting side fell to 93-6.
Gilchrist's brisk 74-ball knock, including seven fours and two sixes, was the lone bright spot for Australia after the early collapse.
The slide started when opener Matthew Hayden (six) was trapped leg-before by Mashrafe bin Mortaza. Captain Ricky Ponting fell similarly to another paceman Shahadat Hossain, the ball keeping low on both occasions.
Earlier, leg-spinner MacGill returned a career-best haul of 8-108, bagging all five remaining wickets on Monday to restrict Bangladesh after they resumed on 355-5.
Bangladesh, who have struggled against most teams in the past, crossed 400 in Tests for only the fourth time. It was their second best total after 488 against Zimbabwe in Chittagong last year during their only Test win.
Bangladesh's total threatened to inflate further after they lost just one wicket in the morning session, but MacGill bowled a superbly tight line and length to run through the batting, conceding just 26 runs.
His haul included Rajin Saleh (67), Khaled Mashud (17) and Mohammad Rafique (six) and bettered his previous best of 7-50 against England at Sydney in 1999.
It was MacGill's fourth five-for against Bangladesh in five innings. He had finished with 17 scalps in their only previous series of two Tests, both of which Bangladesh lost by innings margins.
Warne, who did not play in that series, was again out of action with a sore shoulder and did not bowl on Monday.
"I feel sorry for Warne but am satisfied with my performance. I am tired but it was good to be among the wickets," said MacGill.
"I would have been happier if we were in a better position. The pitch has deteriorated a little but full credit to the Bangladeshi spinners for dominating our batsmen," he added.
Earlier, Bangladesh's overnight pair of Saleh and Mashud saw off the threat of the second new ball, carefully ignoring deliveries outside the off stump.
Saleh, who completed a useful fifth half-century in 16 Tests, was lucky to survive a chance on 48 when paceman Jason Gillespie dropped an easy return catch.
The second and final Test will be held in Chittagong from April 16.