After the tourists fielded well to restrict Australia to 198 for five from their 50 overs, they were unable to capitalise on a fine half-century from opener Tamim Iqbal and were dismissed for just 125 in 29.5 overs in reply.
The 73-run win meant Australia completed a clean sweep in the three-match series, in which the Bangladesh's batting was well below the standard required of an international side.
In the first match they were dismissed for just 74 and lost by 180 runs, then they were bowled out for 117 in the second match on their way to an eight-wicket defeat.
Bangladesh were again in big trouble early in the chase Saturday, slumping to 22 for three before Iqbal and Shakib Al Hasan, who had also done well with the ball, steadied the ship.
The pair put on 60 to lift the visitors back into the match, but when the latter fell for 27 the rot set in again.
Medium pacer James Hopes (3-30) conceded 11 runs from his first over but bounced back to take the vital wicket of Al Hasan and then two more quick scalps as Bangladesh lost 5-21 to slump to 8-103 as all hope was lost.
Man-of-the-match Iqbal eventually fell for 63 from just 69 balls, holing out to Shane Watson (2-8) when he knew he was running out of partners.
Their cause was not helped by the fact that they without number 11 Shahadat Hossain, who picked up a hand injury in the field.
Earlier, on a slow wicket that suited the Bangladesh spinners, the tourists did well to restrict the home side after Australia won the toss and elected to bat.
Mike Hussey again anchored the Australian effort with an unbeaten 57 from 72 balls, while Cameron White added some late runs with 22 not out from 24.
The Australians had looked set to fall well short of 200, until a late flurry from the pair added 23 runs from the last two overs.
The bowlers were able to restrict the strokeplay of the normally aggressive Australians and were superbly backed by their fielders, with three run outs robbing the home side of any momentum.
Spinners Al Hasan (0-25 from 10 overs) and Abdur Razzak (1-22 from 10) both bowled superbly in the conditions.
Opener Watson (27), struggling captain Michael Clarke (25 from 66 balls) and middle order batsman David Hussey (11) were all run out, the latter two both by direct hits at the bowler's end as they attempted quick singles.
The home side went 27 overs without a boundary before Mike Hussey finally broke the drought in the 42nd over.
Bangladesh made two changes to their side for Saturday's match, with Mahmudullah and Farhad Reza replacing Mehrab Hossain jnr and Nazmule Hossain.
Reza's introduction into the attack for the first time in the series was successful, bowling opener Shaun Marsh (30), who had already been dropped, with his first ball.