Michael Hussey's century and another Shane Watson blitz set Australia on the path to their fourth-highest ODI total of all time, and while the final margin of 66 runs may appear comfortable, Bangladesh put up a feisty chase.
The hosts needed 362 for victory, which even their most ardent fans must have felt was unachievable after they managed only 210 and 229 in the first two matches.
But Imrul Kayes and his top-order colleagues gave the Mirpur crowd something to cheer about, pushing the score to 179 for 1 with 30 overs remaining, and Clarke was scratching his head for an answer.
It came in the form of the debutant fast bowler James Pattinson, who picked up his first wicket for his country - not the same country his brother Darren represented - when Kayes edged behind for 93, and the required run-rate crept into unrealistic territory.
Earlier, Michael Hussey hit an impressive hundred and Shane Watson made a quickfire fifty as Australia posted 361-8.
Hussey cracked two sixes and nine fours in a 91-ball 108 for his third one-day hundred, and his first since February 2007, while opener Watson smashed three sixes and 11 fours in his 40-ball 72.
Middle-order batsman Hussey was involved in two useful stands, adding 89 for the fourth wicket with skipper Michael Clarke (47) and 70 for the sixth wicket with Mitchell Johnson, who hammered 41 off just 24 balls.
Hussey fell in the last over, trapped lbw by seamer Shafiul Islam, but not before boosting his team's chances of sweeping the series. Australia already hold an unbeatable 2-0 lead.
Watson, who hit a record 15 sixes in his career-best 185 not out in the previous match on Monday, put the Bangladeshi attack to the sword again with his exciting strokeplay.
Australia got off to a blazing start after winning the toss as Watson and make-shift opener Ricky Ponting (47) added 110 runs for the opening wicket in just 11.5 overs.
Watson began aggressively, hitting Shafiul for four boundaries in an over early in his innings before pulling left-arm spinner Shakib Al Hasan over mid-wicket for a six to reach his half-century.
The Australian opener, who completed his fifty off just 25 balls with the help of two sixes and nine fours, was bowled in the 12th over while attempting to sweep left-arm spinner Abdur Razzak.
Ponting also played attractive shots during his 50-ball knock, hitting two fours and a six from a Mashrafe Mortaza over. He looked well set before being trapped lbw by Razzak.
Razzak finished with 3-58 and Mortaza with 3-80. Australia rested fast bowler Brett Lee and wicketkeeper-batsman Brad Haddin, bringing in Tim Paine and paceman James Pattinson, who made his one-day debut.
Bangladesh replaced Raqibul Hasan and Rubel Hossain with Mortaza and Alok Kapali.
Brief scores
Australia:- 361 for 8 (Hussey 108, Watson 72, Razzak 3-58, Mortaza 3-80)
Bangladesh:- 295 for 6 (Kayes 93, Mahmudullah 68*, Nafees 60, Johnson 3-67)
Results:- Australia won by 66 runs
MOM:- Miachel Hussey (Australia)
Man of the series:- Shane Watson (Australia
