Both teams were looking to bounce back from losing their opening Group Two games and Australia skipper Steve Smith chose to field after winning the toss.
Smith's men picked up wickets at regular intervals to restrict Bangladesh to a modest 156 for five despite a late flurry by Mahmudullah who hit seven fours in an unbeaten 49.
Leg-spinner Adam Zampa (3-23) and all-rounder Shane Watson (2-31) were the pick of the bowlers.
Zampa claimed the key wicket of Shakib Al Hasan who had notched a patient 33 off 25 balls and was just starting to open up.
Openers Khawaja (58) and Watson (21) then got Australia off to a good start, sharing a stand of 62 in 44 deliveries before the latter was run out.
Khawaja was bowled by Al-Amin Hossain in the 14th over having struck seven fours and a six in a 45-ball knock that gave Australia a solid platform, although they had the same sort of middle-order slump they suffered in the first game against New Zealand.
Smith, out for six against the Kiwis, was bowled by Mustafizur Rahman for 14.
David Warner (17) was next to fall and was soon followed by Mitchell Marsh (six).
MAXWELL STUMPED
Glenn Maxwell then scored a brisk 26 before being stumped while all-rounder John Hastings, brought in by Australia for left-arm spinner Ashton Agar, went for three when he was caught at mid-wicket by Soumya Sarkar.
"It's something we spoke about recently, the middle overs...we didn't do well against New Zealand and we haven't done it well tonight either," Zampa told reporters.
"(But) when you're chasing a total where you don't have that many runs to get, you can probably afford to take a few risks."
Australia needed five runs off the last two overs and James Faulkner and Peter Nevill saw them through to a win that lifted the team to third in the group behind New Zealand and Pakistan but ahead of India.
Bangladesh, who are bottom, have a mountain to climb to reach the semi-finals.
"I know this is a high-scoring ground," captain Mashrafe Mortaza said. "We made a few mistakes but I think the way we played we can take a lot of positives into our next games."
(Editing by Tony Jimenez)
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