Clarke had led by example with his 101 as Australia posted 270 for seven after being put in by Bangladesh.
Clarke, who was confirmed as Ricky Ponting's replacement at the end of last month, reached his century off 110 deliveries, an innings including six fours and two sixes.
Bangladesh struggled to step up the tempo in their reply and could only coast to 210 fo five, Tamim Iqbal top-scoring with 62.
After Clarke had lost the toss and seen his side put in, they lost an early wicket when Brad Haddin was out in just the fourth over for 10 to be replaced by Ponting.
The former skipper produced some vintage shots as he and Watson put on 65, with Ponting hitting five fours and one six before being run out for 34 off 31 deliveries.
Clarke joined Watson at the crease but the latter was out with just nine more on the board when he was lbw to Abdur Razzak for 37 off 32.
Cameron White and Clarke put on 48 before White was caught by Kayes after a weak innings of just 20 off 50 deliveries.
Mike Hussey (33) then came to the crease and helped steer the tourists to 222 before becoming Shuvo's second victim with Mahmudullah taking a simple catch.
Shuvo quickly struck again with just three more runs on the board, bowling Steven Smith for one to leave the visitors on 225 for six after 46 overs.
Clarke reached his ton with a huge six off Mortaza over deep midwicket but holed out to Mahmudullah next ball.
The final ball of the 50 overs saw Mitchell Johnson hit a huge six, again off Mortaza, as he finished unbeaten on 26 off 13, with two boundaries and two sixes.
Australia then had a dream start with the ball, reducing their hosts to nine for two.
Brett Lee struck first, Imrul Kayes playing at the Australian's full delivery and feathering to wicketkeeper Haddin for four.
With just one more run on the board, Bangladesh lost another wicket as Shahriar Nafees was out lbw to Mitchell Johnson first ball.
The hosts were wobbling but Tamim and Raqibul Hasan brought some stability, adding 40 before the latter was removed for seven off the bowling of John Hastings.
Shane Watson, second top scorer for Australia, produced a diving catch to his left at short extra cover from Raqibul's drive to leave Bangladesh on 49 for three.
The onus was now on Tamim and Shakib Al Hasan to steer the hosts and the pair shared a stand of 67 before the former was out, silencing the crowd.
Three deliveries after slogging a six over midwicket off Xavier Doherty, Tamim attempted a repeat but instead he picked out Smith, who held a catch low down.
Tamim's innings of 62 came off 89 deliveries and included four fours and that one six.
With the hosts ticking along nicely, Shakib reached his half-century before charging down the wicket to a Smith delivery which he got under and hoisted into the air for Watson to take an easy second catch of the day.
Shakib made 51 off 90 deliveries with just three boundaries.
Mushfiqur Rahim finished 45 not out and shared in an unbroken stand of 61 at better than a run a ball with Mahmudullah (28no), but the Tigers still came up well short.