Clarke, was appointed as Australia's Test and ODI after Ricky Ponting resigned from his post after the World Cup. Clarke scored 101 off 111 balls.
Clarke's knock ensured that Austraia ended at 270-7. Bangladesh in reply lost early wickets and slipped to 49-3.
Only vice-captain Tamim Iqbal (62) and captain Shakib Al Hasan (51) offered any resistance, but the home team finished well short on 210-5.
Earlier, Australia, who lost to India in the World Cup quarter-finals, were 138-4 in the 30th over before Clarke propped up the innings with a vital 84-run stand for the fifth wicket with Michael Hussey (33).
Clarke, needing five runs off the last three deliveries of the innings to complete his sixth one-day hundred, smashed paceman Mashrafe Mortaza for a six over mid-wicket before being caught in the deep off the next ball.
He hit two sixes and six fours in his 111-ball knock.
Left-arm spinner Suhrawadi Shuvo was the most successful bowler with three wickets for 44 runs, including those of Hussey and Steven Smith (one) in the same over.
Bangladesh, electing to field after winning the toss on a slow pitch, tasted success in the fourth over when Mortaza, who missed the World Cup due to a knee injury, bowled Brad Haddin for 10.
Opener Shane Watson (37) and Ponting (34) then dominated the attack with some handsome shots. The opener hit Mortaza for a four and a six off successive balls before lofting left-arm spinner Abdur Razzak over mid-wicket for a six.
Ponting also looked in excellent touch, smashing three successive fours off Mortaza and hoisting Shuvo over long-on for a six. His brilliant cameo ended when he was run out going for a third run.
Watson, who added 65 for the second wicket with Ponting off just eight overs, also fell when looking well set as he was trapped lbw by Razzak after hitting two sixes and three fours in his 32-ball knock.
The remaining two one-dayers will be played in Dhaka on 11 and 13 April.
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