Ponting made 101 from 118 balls while Bevan crafted 57 at exactly a run a ball as the world champions look to seal a 3-0 series triumph at the Marrara Oval following two easy victories in Cairns.
Ponting and Bevan also shared a 127-run partnership for the fifth wicket to put the hosts back in control after the middle-order collapsed, losing three wickets for two runs in 14 balls.
After winning the toss and electing to bat first, Australia made a great start with Adam Gilchrist and Matthew Hayden putting on 54 for the opening wicket before Gilchrist fell to spinner Mohammad Rafique for 31.
Rafique captured a second wicket when he took a return catch to get rid of Hayden for 42, triggering a minor collapse.
Damien Martyn departed for one when he played on to Alok Kapali then Andrew Symonds was run out for a duck as the Australians slumped to 114-4 before Ponting and Bevan steadied the ship.
Ponting made a slow start on a pitch that was proving difficult to score off but quickly grew in confidence as his innings progressed, smashing four sixes and two fours to post his 14th one-day international ton, before being dismissed in the 49th over.
Bevan also wielded the long handle, cracking two fours and two sixes, before he too threw his wicket away in the frantic final few overs, succumbing to a full toss from Mashrafe Mortaza, who also dismissed Ian Harvey for five on the last ball of the innings.
