Bevan was named as man-of-the-match with his match double of 2 for 26 and 42 as Australia overtook the Kiwis' 141 in the 39th over at 142 for 4.
Ponting was unbeaten on 60, scored off 92 balls, to put the Australians into second place in the triangular series behind the leaders South Africa heading into the Christmas recess.
Ponting and Bevan put on a match-winning 95-run partnership for the fourth wicket. When Bevan was out to become Chris Cairns' fourth wicket, Australia were 117 for 4 and 25 runs from victory with the luxury of 18 overs left.
Yet again Australia's one-day captain, Steve Waugh, had a poor match, out for a second-ball duck to continue his run of failures after taking over the captaincy for the limited-overs international series from Mark Taylor, who remains as Test skipper.
Waugh was out for a duck for the second time in as many games when he edged the second ball he faced from Cairns to the wicketkeeper Adam Parore.
Earlier, Chris Harris, with an undefeated 62, steered New Zealand from a calamitous position to be all out for 141 with three balls left of their 50 overs. At one stage New Zealand were 75 for 7.
South Africa Students took advantage of some generous fielding by England Under-19s to reach 255 for 4 at the close of the first day of their three- day fixture in Port Elizabeth.
India and New Zealand were involved in a tied women's World Cup match in Indore yesterday. Both sides had already qualified for the quarter- finals from Group B, but still went all out for victory. Chasing a target of 177 in the stipulated 50 overs, India slipped from 121 for 2 to 176 all out in the final over.