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, briefly threatening an upset when they reached 82-3 on the back of a splendid half-century from Tamim Iqbal.
The teenage opener made 63 off 69 balls, featuring five fours and a six, but received little support from his team mates as only two other players made double figures. He threw his wicket away when he started running out of partners.
"I played a stupid shot, I shouldn't have played that shot and should have carried on the innings," he said. "The result would be different if I'd carried out the innings."
Bangladesh coach Jamie Siddons was pleased with the progress of his bowlers but said the batting needed drastic improvement to challenge the best teams.
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