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Leg-spinner Adam Zampa bowled the Aussies to a resounding eight-wicket victory in Dubai on Thursday.
Australia and South Africa are vying to grab the second semi-final spot from Group 1, behind leaders England, and Aaron Finch's team needed a net run-rate boost to go with a victory and their bowlers rose to the occasion.
Zampa (5-19) led the rout, skittling out Bangladesh for 73 in 15 overs. Finch smashed 40 off 20 balls as Australia romped home in 6.2 overs for their third win in four matches, while Bangladesh ended their Super 12 campaign losing all five matches.
"Really clinical performance," Finch said after Australia leapfrogged South Africa into second place on account of a better net run-rate though both are level on six points. "We thought we would take the opportunity to win big if we got the chance, but you can't plan for that."
Australia face West Indies in their final Super 12 match on Saturday when South Africa play England.
The Windies are out of the T20 World Cup after a heavy defeat to Sri Lanka in Abu Dhabi.
Sri Lanka’s Charith Asalanka and Pathum Nissanka hit 68 and 51 respectively, both off 41 balls, to create a run chase of 190 for West Indies, which was enough to earn a second victory in their final match of the tournament.
Chris Gayle was out early for one and Shimron Hetmyer’s spirited half-century was not enough for a side which fell 20 runs short.
Additional reporting by Reuters.