The World champions posted a modest total of 198 for five from their 50 overs but sealed a comfortable victory by bowling out Bangladesh for 125 in the 30th over.
Mike Hussey, who was named player of the series, top-scored for the Australians with an unbeaten 57 while opener Shaun Marsh made 30 after scoring half-centuries in the previous two matches.
"It was closer than the first two games but our mindset was to win this game," Australia's acting captain Michael Clarke said. "Bangladesh batted well at the start, they were positive, but we all knew if we could just take one or two wickets, it would put them on the back foot."
Bangladesh, who were thrashed in the two previous matches, produced easily their best performance of the series, but that wasn’t enough to beat the Australia.
SCORE BOARD
AUSTRALIA
Marsh b Reza 30
Watson (run out) 27
Clarke (run out) 25
M Hussey (not out) 57
Hussey (run out) 11
Haddin c Kapali b Razzak 16
White (not out) 22
Extras (LB-1, W-9) 10
Total (for 5 wkts, 50 overs) 198
Fall of wickets: 1-41, 2-84, 3-87, 4-111, 5-150.
Bowling: Mortaza 10-0-55-0, Shahadat 6-0-37-0, Reza 10-0-43-1, Saqibul 10-0-25-0, Razzak 10-1-22-1, Riyad 4-0-15-0.
BANGLADESH
Iqbal c Clark b Watson 63
Siddique c White b Clark 0
Ashraful c Clarke b Clark 3
Kapali b Johnson 0
Saqibul c M Hussey b Hopes 27
Riyad b Hopes 1
Dhiman lbw Hopes 1
Reza c D Hussey b Watson 0
Razzak b D Hussey 14
Mortaza (not out) 8
Shahadat (absent hurt)
Extras (B-2, LB-2, W-4) 8
Total (all out, 29.5 overs) 125
Fall of wickets: 1-2, 2-13, 3-22, 4-82, 5-96, 6-98, 7-99, 8-103, 9-125. Bowling: Bracken 6-1-19-0, Clark 5-1-12-2, Johnson 4-0-36-1, D Hussey 4.5-0-16-1, Hopes 5-1-30-3, Watson 5-0-8-2.