Set 291 to win, the Australians looked in deep trouble at 3-112, but man-of-the-match Ponting and Brad Hodge, who finished on 99 not out, came together and led their side in a record run-chase at the ground, winning with 10 balls to spare on Sunday.
In reply to New Zealand's 290 for seven, Australia finished on 291 for five as the Black Caps' frustrating run of narrow losses to the Australians continued.
The Australians have now won 20 of their past 22 matches against their near neighbours.
Ponting and Hodge put on 154 for the fourth wicket to turn on its head a game that appeared at New Zealand's mercy.
The skipper eventually fell for 104, made from 113 balls with six fours and one six, but Hodge guided the Australians home.
While Ponting, back from a hip injury, continued his imperious form and did not give a genuine chance until he was caught, Hodge was the surprise packet as he played for his career on his home ground.
Given another chance at international level only because of the injury to Andrew Symonds, Hodge was on his last chance and started nervously.
However, he blossomed as his partnership with Ponting developed and played himself into World Cup contention.
He and Ponting needed more than eight runs an over at one stage, but remained calm under pressure as the Kiwis started to wilt and lost control of their bowling.
Hodge's first 17 runs were all singles, but then he started to find the boundary and faced just 86 balls, hitting seven fours and one six and fittingly scoring the winning runs.
The result means the winner of Tuesday's clash between New Zealand and England will face Australia in the tri-series finals.
The run chase came after Black Caps opener Lou Vincent backed up his tough talk with a superb 90 to lay the foundation for New Zealand's total.
Having labelled the Australians arrogant in the media earlier in the week, Vincent was under pressure to deliver and did so despite a searing spell of fast bowling from Australian quick Shaun Tait.
Tait eventually got his man with the penultimate delivery of his 10 overs, but Vincent had already done enough to ensure the Black Caps would set Australia a big target under lights after captain Stephen Fleming won the toss and elected to bat.
Vincent combined with the busy Peter Fulton (60) for a 151-run second-wicket partnership.
It was the rejuvenated Vincent's third half-century in as many matches since being recalled to replace the retired Nathan Astle.
Vincent looked set for a century until Tait burst through his defences and bowled him off his pads.
The wicket capped a fine bowling performance from Tait, who copped severe treatment in his first game but may have bowled himself into the Australian World Cup squad.
In just his second One-day international, he claimed 1-26 from his 10 overs and stepped into the rarefied group of the quickest bowlers in the world.
In his second over, Tait sent down a delivery to Vincent that clocked 160 kilometres an hour (99 miles an hour), just shy of the magical 100 mph (160.9 kmh) barrier.
In doing so, he became just the third bowler, behind Pakistan's Shoaib Akhtar and his opening partner Brett Lee, to officially reach 160 kmh.
At the other end of the scale, some assured spin bowling by Michael Clarke helped rein in the Black Caps, who looked set to score more than 300 when they were 1-162 after 31 overs.
Clarke (2-45) claimed the wickets of Fulton and Ross Taylor with his well-flighted finger spinners.
The Black Caps hit 50 off the last four overs as Australia's recent problems containing sides in the final overs continued.