They were lucky to avoid an embarrassing defeat in Fatullah when captain Ricky Ponting dug in to score an unbeaten century, steering his side narrow three-wicket victory in the opening test of the two-match series.
Tired after their tour of South Africa, Australia were in desperate trouble after the first two days when Bangladesh scored 427 in their first innings before dismissing the visitors for 269.
The Australians managed to take out the win after their attack came alive in the second innings to bowl out the hosts for 148, leaving them to score 307 for victory.
Australia's great escape in Fatullah snuffed out Bangladesh's hopes of recording only their second victory in 43 tests.
Instead of chalking up one of the most sensational upsets in test history, the hosts were condemned to their 38th defeat.
Injury and fatigue, however, remains a major concern for the visitors.
''We have to look at how Shane (Warne) and the guys, who had a heavy workload in South Africa feel now,'' Ponting told reporters.
''I am worried about a couple of fast bowlers in particular with the workload they had in South Africa.
''We have to sit down and talk what might do, what is more important to me is to give the guys time-off to really get back into their game at the start of the second test.'' Mental Challenges Australia have called left-arm seamer Nathan Bracken and off-spinner Dan Cullen into their squad.
''We have to be ready physically and mentally for the challenges we are facing in Chittagong,'' Ponting said.
Despite their loss, Bangladesh should be boosted from the progress they have made since their last outing against the world's most dominant side.
They suffered innings defeats in two previous tests during a tour of Australia in 2003, the first of which lasted only 2-1/2 days.
Bangladesh will now want to keep up the momentum they have gained.
''We are very happy with our performance and we want to improve it further in the second test,'' Bangladesh captain Habibul Bashar said.
''We know that the Aussies will try to come hard on us in the second test but my past experience says we are going to have another good match in Chittagong.'' Bangladesh's only test victory to date was against a depleted Zimbabwe last year.