New Zealand, chasing 244 to win, were on 51 for two off 15 overs when rain stopped play in the Group A clash at 3.40pm local time (1440GMT).
Under the Duckworth/Lewis system for rain-affected matches, a minimum of 20 overs had to be bowled in the second innings for either side to win this fixture.
But at 6.25pm (1725GMT), the umpires decided no further play was possible.
The first no result of the tournament saw both sides take a point apiece.
That left New Zealand with three points after their one-wicket win over Sri Lanka and gave Australia their first point of this event after a 48-run loss to hosts England at Edgbaston last week.
Adam Voges’ top-scored with 71 in a total of 243 for eight with George Bailey, Australia’s stand-in captain, making 55 after winning the toss. New Zealand left-arm quick Mitchell McClenaghan took four for 65.
Brief scores: Australia 243-8 from 50 overs (Voges 71, Bailey 55, McClenaghan 4-65) New Zealand 51-2 from 15 overs. (Abandoned)