The left-handed opener smashed 58 off 45 balls as Australia successfully chased down Bangladesh’s total of 156 for five.
The victory puts Australia’s bid for a first ever World T20 trophy back on track after the top-ranked Test side lost to New Zealand in their opening match.
The 29-year-old Khawaja whacked seven fours and one six before he was bowled by Al-Amin Hossain in their Super 10 Group Two match.
Khawaja, born in Pakistan, and Shane Watson put on a solid 62-run opening partnership before the veteran all-rounder was run out for 21.
Bangladesh’s string of bowlers mounted a spirited attack against the Aussies, after suffering a devastating blow when Taskin Ahmed and Arafat Sunny were suspended for illegal bowling actions at the weekend.
Skipper Steve Smith was bowled for 14 by Mustafizur Rahman and David Warner was caught and bowled for 17 by Shakib Al Hasan. Glenn Maxwell was stumped for 26 off 15 balls.
James Faulkner hit the winning runs with nine balls to spare.
Bangladesh, demolished by Pakistan in their first group match, are now unlikely to qualify for the semi-finals, sitting at the bottom of their group with two matches left to play.
After Australia won the toss and elected to field, Mahmudullah top scored for the Tigers, striking an aggressive unbeaten 49, that included seven fours and one six.
Seasoned T20 campaigner Shakib also chipped in with 33 off 25 balls, as Bangladesh desperately tried 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 Watson claimed two.
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 Maxwell to take a strightforward catch at point.
Sabbir Rahman scored 12 off 17 balls before 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 and Sunny, was lbw to Zampa for 13.
Bangladesh now play India on Wednesday also in Bangalore, while Australia take on Pakistan on Friday in Mohali.