Australia's collapse came in the final 15 overs of a session they had dominated with leg-spinner Stuart MacGill enjoying career-best figures of 8-108 to polish off the Bangladesh innings after they had reached lunch on the second day at 413-6.
At the interval, Mike Hussey was the not out batsman on 19 after Damien Martyn (4) was bowled by spinner Mohammad Rafique with the fourth ball of the final over before tea.
Paceman Mashrafe Mortaza had earlier given the Bangladesh attack a perfect start when he trapped opener Matthew Hayden lbw for six with the second ball of his second over to leave Australia on 6-1.
Skipper Ricky Ponting was the next man out with the score on 43, when he was also trapped lbw by fast bowler Shahadat Hossain for 21.
The triple strike represents a brilliant response from the home side after Australia had hauled themselves back into the contest on the second afternoon.
In just 8.3 overs following lunch, MacGill bowled Rafique for six, caught and bowled Rajin Saleh, trapped Hossain lbw for three and had Enamul Haque Jr caught by Matthew Hayden for a duck to wrap up the innings.
However, the home side will have been more than delighted with their second-highest total in a Test innings that was inspired by a maiden Test century from Shahriar Nafees (138) and fifties from skipper Habibul Bashar (76) and Saleh (67) Before the late slump. The home side had looked solid in the morning session, adding 58 runs to their overnight 355-5 for the loss of Khaled Mashud (17).
MacGill took the only wicket of the morning when he lured Mashud forward out of his crease and keeper Adam Gilchrist whipped off the bails after the batsman had played and missed.
Saleh almost missed out on his fifth Test half-century when Jason Gillespie dropped a caught and bowled chance with the batsman on 48, but he has otherwise looked solid and occupied the crease for 203 deliveries in total, hitting six boundaries.
Australia's Shane Warne did not take to the field this morning, the leg spinner complained of a sore shoulder the previous day after bowling 20 overs for 112 runs and no wickets.
The second of the two-match series starts in Chittagong on Sunday.