Taskin Ahmed had the recalled opener Matt Renshaw caught for eight in the slips.
Bangladesh were bowled out after lunch, after Australia captain Pat Cummins won the toss and elected to field on a bouncy wicket prepared like a typical Gabba pitch.
Australia paceman Mitchell Starc took five wickets in 22 balls. The left-armer finished with 6-12 after completing the sixth-fastest five-wicket haul in tests. Starc holds the record for the fastest (15 balls) against West Indies in Kingston last year.
Starc struck with the first ball of the match to remove opener Shadman Islam and captured the first five wickets, leaving the tourists reeling at 12 for five after 6.4 overs. He returned to have tailender Shoriful caught behind for 14 to wrap up the innings.
Apart from Shadman, opener Tanzid Hasan, captain Najmul Hossain Shanto and wicketkeeper Litton Das all fell for ducks off Starc's bowling.
Shadman was caught in the gully after sending a thick edge flying to Cameron Green. Tanzid Hasan drove loosely at a fuller ball to nick to Beau Webster in the slips, while Shanto was trapped lbw on the next ball.