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).
