The Aussies managed to restrict Bangladesh to 156-5 from their 20 overs, thanks to three wickets from young spinner Adam Zampa.
In reply, Australia finally got over the line by three wickets, after a mini-collapse had occured following opener Usman Khawaja's 58.
The victory keeps Australia's hopes of qualification alive, whilst Bangladesh are all-but out following two defeats from their opening two matches in Group 2 of the Super 10 stage.
Shakib holds innings together after Watson strikes early with double blow
After Steve Smith had won the toss and chosen to bowl first, Shane Watson stepped up for his skipper knocking over two Bangladeshi batsman, justifying his captain's decision.
First to go was Soumya Sarkar, whose poor form continued when he was caught for one by Glenn Maxwell after mistiming a drive off Watson.
Next to fall to Watson in the sixth over was Shabbir Rahman, who was well caught by James Faulkner for 12 after he had made a solid start.
Shakib Al Hasan then came to the crease, to join Mohammad Mithun, who had opened in place of the ill, Tamim Iqbal. The pair added 37 in four overs to give the innings some impetus before Mithun was caught by Watson off the bowling of leg-spinner Adam Zampa.
Mahmudullah and Rahim smash Tigers to competitive total
Zampa continued to impress for the Aussies, picking up two more wickets in the middle overs. Firstly he saw off debutant Shuvagata Hom, who was trapped in front for 13, which was then followed four overs later by the big wicket of Shakib who was caught for 33 by Nathan Coulter-Nile.
At the fall of Shakib's wicket, Bangladesh were 105-5 with 28 balls remaining in the innings, and looked set to post a total in the region of 130-135. However, some poor bowling coinciding with some great batting, saw Bangladesh eventually post 156 from their 20 overs.
Mahmudullah was the main man for the Tigers, finishing unbeaten on 49 from 29 deliveries, which included seven boundaries and a six. At the other end, the diminuative Mushfiqur Rahim played his usual array of deflective shots to finish 15 not out from 11 balls.
Khawaja starts chase well, making 58
With calls for Aaron Finch to return to the side, Usman Khawaja continued his fine form, making 58 from 45 balls opening-up.
The left-handed played some beautfiul strokes in the powerplay, completely dominating the partnership with Watson. That partnership reached 62 before the first wicket fell, with Watson run-out for 21 after a strong throw from the boundary by Shabbir.
Steve Smith was the next batman to come-and-go, when his short stay at the crease was bought to an end for 14 by Mustafizur Rahman.
Australians limp across the line, to claim crucial win
With 42 runs needed from the final seven overs with eight wickets in hand, it looked as though Australia would win quite comfortably and improve their run-rate.
That however was not the case, with the Australian middle-order suffering another collapse which nearly cost them dearly.
Khawaja was bowled around his legs by Al-Amin Hossain for 58 to start the rot, with David Warner following in the next over for 17 when he was caught-and-bowled by Shakib.
Mitchell Marsh then perished for six, after he was deceived by Mustafizur, but at the other end Maxwell helped to bring the total down with a trio of fours off of Saqlain Saqib.
With five runs still needed, John Hastings became the next batsman to give his wicket away, smashing a Shakib delivery straight to Sarkar on the boundary.
It was then left to Faulkner to finally hit the winnings runs off of Saqib to give Australia victory with nine balls remaining and kickstart their campaign.