The Australians had earlier dismissed New Zealand for 238 and set a target of 239 for victory.
Australia found their way home with 17 balls to spare and nailed the trophy with a 3-1 series scoreline.
White and Michael Hussey [ Images ] completed the chase, which was set up by Ponting and Shane Watson [ Images ]. And despite a couple of quick wickets in the middle from Daniel Vettori [ Images ], the visitors were on track for most of the evening.
Man of the Match White was cool with an unbeaten 50 and Hussey finished on 28, easing any tension the Australians felt when Adam Voges (34) was caught at midwicket with 50 runs still required.
Ponting was powerful with back-foot drives and pulls, and although he should have been lbw for 19, he reached his fifty from 34 balls before being trapped in front by Vettori from his next delivery.
Watson (32) was also lbw to Vettori, hurried by an arm ball, but in the reduced contest New Zealand needed a bigger run of wickets to have a good chance.
Earlier, New Zealand were dismissed for 238 off just 44.1 overs after being sent into bat by Australia.
After Brendon McCullum [ Images ] got the hosts off to a flying start with 61, the Kiwis' middle order collapsed as they were dismissed inside the 50-over mark for a second consecutive match.
Only a late flurry from No 9 Daryl Tuffey [ Images ], who hit four sixes in his 34 off 17 balls, saved New Zealand from total embarrassment.
All the Australian bowlers chipped in, with spinner Nathan Hauritz [ Images ] taking three for 46 off eight overs after Tuffey hit him for three consecutive sixes.