Batting first at Allan Border Field, Bangladesh were bowled out for 166 in 49.4 overs, with veteran all-rounder Ritu Moni top-scoring with 45 off 63 balls.
Captain Nigar Sultana Joty made 29 off 86 deliveries, while Shorna Akter added 21 off 50 balls.
Australia's Megan Schutt struck with the first ball of the match and claimed two wickets in her opening spell, reaching 150 ODI wickets in the process.
Annabel Sutherland led the attack with four wickets for 30 runs, while Schutt finished with 2-19 from 10 overs.
In reply, opener Phoebe Litchfield smashed 77 off just 57 balls to guide Australia to victory in 25.4 overs. Georgia Voll scored 19, while Beth Mooney made 13 before falling to debutant Habiba Islam Pinky, who claimed two wickets for Bangladesh.
The two teams will meet again at the same venue on Sunday.
