Chasing 285, Australia slumped to 191-9 when rain halted play, with only four batters making it into double figures. Cameron Green top-scored with an unbeaten 52, but he found little support as wickets fell at the other end at regular intervals.
Fast bowler Nahid Rana led Bangladesh’s attack with four wickets from his 10 overs.
Earlier, Bangladesh built a competitive 284-8 after half-centuries from Tanzid Hasan Tamim, Najmul Hossain Shanto and Mosaddek Hossain.
Australia were without captain Mitchell Marsh, who was ruled out of the ODI series with an ankle injury. Spinner Tanveer Sangha is also unavailable for the tour, while opener Travis Head has been rested.
The second ODI will be played on Thursday.