Australia posted 238 runs for the loss of seven wickets in their 50 overs after they lost the toss and were sent in to bat.
New Zealand's reply amounting to 136 petered out in the 42nd over.
Australia were in early trouble when openers Leah Poulton and Shelley Nitschke departed with just 17 runs on the board.
But a 154-run partnership for the third wicket between skipper Alex Blackwell and Sarah Elliott steered the Australians out of trouble before both fell to Aimee Watkins.
Watkins led the New Zealand fightback, taking another two of the four remaining Australian wickets to fall in the space of 18 runs by the 45th over.
But Rachael Haynes and Ellyse Perry stood firm to raise an unbeaten 49-run partnership from the remaining overs.
New Zealand, having lost by 115 runs and by four wickets in the first two matches, were fragile at the crease, and unable to withstand close scrutiny by the Australian bowlers who were backed up by sharp fielding.
From 98 for six by the 30th over, the White Ferns were able to add just another 38 runs before they were all out.
Nitschke was the best of the Australian bowlers taking three for 31 with Lisa Sthalekar and Elliott chiming in with two wickets each.
The remaining matches of the series will be played at Melbourne on Wednesday and Thursday.
