Australia, who have taken a winning 2-0 lead in the series, looked set to complete a whitewash despite fielding a weakened team under Adam Gilchrist, who captained the side after regular skipper Ricky Ponting decided to sit out.
Left-arm wrist-spinner Brad Hogg was pressed into attack and finished with three wickets, while Mitchell Johnson and Daniel Cullen bagged two scalps each.
Opener Rajin Saleh (37) and captain Habibul Bashar (33) were the only ones who came to terms with the pitch and Bangladesh lasted just 42.3 overs after electing to bat.
Saleh's painstaking knock came off 102 deliveries before he was bowled by left-arm spinner Michael Clarke while trying to sweep. Bashar was stumped by Gilchrist as he jumped out of his crease to Hogg.
Earlier, Bangladesh opener Shahriar Nafees (zero) fell leg-before on the third ball of the innings to left-armer Johnson.
Aftab Ahmed (one) and Mohammad Ashraful (13), who were both dropped for the previous match, gave up their wickets cheaply.
Ahmed flashed outside the off stump to be caught by Gilchrist off seamer Brett Dorey while Ashraful glided the ball straight to Daniel Cullen at short mid-wicket off James Hopes as Bangladesh fell to 41-3.
Saleh and Bashar added 39 for the fourth wicket but Bangladesh offered litte els as Hogg, who had bagged three wickets in the previous match, continued his fine run.
Australia won the preceding two-Test series 2-0.