Hussey scored a sparkling 47 not out and Nannes took four wickets as the Aussies ended the first phase of their campaign unbeaten.
The one-day world champions were struggling at 65 for six before Hussey came up with a cameo and shared 74 crucial runs with Steven Smith (27 off 18) for the seventh wicket to lift Australia to 141 for 7.
Hussey took the onus on him to guide Australia's innings and stepped up the tempo with a 17-run over off Mashrafe Mortaza. He hit as many as four fours and six during his 29-ball unbeaten stay.
Nannes then produced a breathtaking display of pace bowling, claiming four wickets for 18 runs as the Bangladeshis were skittled out for 114.
He was well-supported by David Hussey (2-8) and Smith (2-29).
Hussey also caught Mushfiqur Rahim, who had batted with character, off the bowling of his brother David.
Australia, who defeated title-holders Pakistan by 34 runs in their opening match, join Pakistan from Group A in the Super Eight stage and will take on India in their opening second round encounter on Friday.
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