Bangladesh lost opener Misty Shaha on the first ball of the innings but every other batsman kept a cool head in the chase of 131.
Earlier in the day, Australia won the toss and decided to bat first, but the Bangladeshi bowlers reined them in. One-down Claire Moore made 52 but consumed 51 balls while Ella Hayward scored 35 from 39 balls.
Openers Marufa Ektar and Disha Biswas took two wickets each for Bangladesh, while Deepa Khatoon played a key role in containing Australia as she conceded only 16 runs in four overs.