They rode on Michael Hussey's [ Images ] unbeaten 47 and Dirk Nannes's [ Images ] four-wicket haul to finish their group campaign on an unbeaten note, having defeated title-holders Pakistan by 34 runs in their opening match.
Australia join Pakistan from Group A in the Super Eight stage and will take on India [ Images ] in their opening second round encounter on Friday.
Michael Hussey played a sweet little cameo and shared 74 crucial runs with Steven Smith (27 off 18) for the seventh wicket to lift Australia to 141 for 7 before Nannes (4-18) produced a breathtaking display of pace bowling to skittle out the Bangladeshis.
He was well-supported by David Hussey [ Images ] (2-8) and Smith (2-29).
Captain Shakib Al Hasan top-scored for Bangladesh with a run-a-ball 28, while wicketkeeper Mushfiqur Rahim (24 off 25) gave him good support from the other end in a 48-run fifth-wicket partnership.
Bangladesh made a mockery of their run chase as they lost four of their top-order batsmen inside the first four overs.
Needing a outright win, inside 19 overs, to book a place in the Super Eight stage, Bangladesh did not help their cause in any way by playing insensible shots against some fine pace bowling and were tottering at 16 for four.
Shain Tait (1-15)) and Nannes tested the Bangladeshi batters from the onset with their fiery and high quality fast bowling.
Nannes was the quickest of the duo as he picked up Mohammad Ashraful [ Images ], Aftab Ahmed and Mahmudullah in his three-over opening spell before returning to account for the wicket Jahurul Islam (18).
Inspite of the early jitters, Shakib and Mushfiqur put up a brave fight during their the fifth wicket to keep Bangladesh in the hunt.
Shakib's innings was finally cut short by a blinder of a catch at deep by Michael Hussey off Smith.
Jahurul and Shafiul Islam (16 off 13) tried their level best to dig something out of the game but it was not Bangladesh's day as Australia eventually romped home with eight balls to spare.
Earlier, opting to bat, the Aussies initially struggled as they lost six wickets for mere 65 runs before Hussey and Smith saved the day with the 42-ball vital partnership.
After the initial hiccup, the Aussies added 59 runs in the last five overs, courtesy some fine hitting from Hussey and Smith.
Watson mistimed a pull of Mortaza's well-directed bouncer that landed in the hands of Ahmed.
Captain Shakib (2-24) then ensured double celebration for Bangladesh in the next over, picking dangerman David Warner (16 off 11), caught by Ahmed.
A few overs later, a reckless shot from Brad Haddin [ Images ] ensured further trouble for Australia. He was easily caught by Shafiul off left-arm spinner Abdur Razzak (1-29) as he went for a lofted sweep over square leg.
As if that was not enough, skipper Michael Clarke [ Images ] (16) was the next to depart, caught at long-off boundary by Jahurul off Ashraful (1-24). Australia were stuttering at 54 for four at the halfway mark.
It could have been worse had Jahurul not dropped a David Hussey sitter off Ashraful's next ball.
But Mortaza ensured that Bangladesh did not have to pay heavily for the dropped chance by dismissing David Hussey in the next over to mark the departure of half of the Aussie side.
Another well-directed bouncer from Mortaza hurried David Hussey into the pull, but the batsman could not control the shot and ballooned it to wicketkeeper Mushfiqur.
Cameron White [ Images ] too failed to make any impact as Shakib outsmarted him with a lovely leg-spinner that turned away and crashed on to his timbers.
With the chips down, Michael Hussey and Smith first played sensibly, scoring ones and twos, before opening up in the final five overs to take Australia to a fighting total.
Michael Hussey took the onus on him to guide the Australia innings and stepped up the tempo with a 17-run over off Mortaza. He hit as many as four fours and six during his 29-ball unbeaten stay.
Smith too was not far behind as he used his long handle to great affect in the dying stages to register two sixes and a boundary against his name.