Earlier, Mahmudullah struck an aggressive unbeaten 49 as Bangladesh made 156 for five from their 20 overs.
The all-rounder smashed seven fours and one six off just 29 balls after Australian captain Steve Smith won the toss and opted to bowl first in the Super 10 Group Two match.
Seasoned T20 campaigner Shakib Al Hasan also chipped in with 33 off 25 balls, as the Tigers desperately try to notch up their first points in the group stage.
Shakib struck three fours and one six before being caught by Nathan Coulter-Nile off Adam Zampa for the fifth wicket.
Spinner Zampa, 23, spearheaded the Australian bowling attack by taking three wickets, his first in T20 internationals, while veteran allrounder Shane Watson claimed two.
Bangladesh, demolished by Pakistan in their first group match, are reeling after two of their top bowlers were suspended for illegal actions.
Right-hander Mohammad Mithun struck a tidy 23 off 22 balls before being caught by Watson at deep midwicket off Zampa.
Opener Soumya Sarkar was caught for one trying to slash a Watson delivery, only for Glenn Maxwell to take a strightforward catch at point.
Sabbir Rahman scored 12 off 17 balls before James Faulkner took a diving catch off a Watson delivery in the sixth over at the Chinnaswamy Stadium.
Shuvagata Hom, rushed into the side as one of the replacements for Taskin Ahmed and Arafat Sunny, was lbw to Zampa for 13.- AFP.