This was an identical outcome to when the teams met at the Birmingham ground in the last edition four years ago.
Australia, who were left needing a revised target of 235 to win in 33 overs, were 53 for three off nine when the players left the field for the final time.
A minimum of 20 overs had to be bowled in Australia's innings for a result to be declared but the last downpour was so severe there was no chance of the match resuming in time.
The teams took a point apiece – a result that was arguably of most benefit to Group A rivals England, who launched a tournament featuring the world's top eight one-day international sides with an eight-wicket win over Bangladesh at the Oval on Thursday.
Only the top two sides from each of the two groups of four qualify for the semi-finals.
New Zealand were on course for a score well above 300 while Williamson was at the crease in an innings reduced from the ODI standard 50 to 46 because of rain.
But his run out sparked a slump that saw the Blackcaps dismissed for 291 after losing their last seven wickets for 37 runs in 35 balls.
Josh Hazlewood was the chief beneficiary as several batsmen carelessly holed out, with the fast bowler returning ODI best figures of six for 52.
Hazlewood took three wickets in four balls to wrap up the innings with an over to spare.
But further rain during the mid-innings break left Australia with a tough new target under the Duckworth-Lewis-Stern method for weather affected limited overs matches.
Australia, trying to force the pace against accurate new-ball bowling, were soon 35 for two after big-hitting openers David Warner (18) and Aaron Finch (eight) fell cheaply.
Warner, advancing down the pitch to cut left-armer Trent Boult was caught behind before Finch clipped Adam Milne to mid-wicket.
And when Moises Henriques (18) was caught and bowled by Milne via an inside edge off his pad, Australia were three down.
But no sooner had Henriques been dismissed then fresh rain took the players off the field.
Australia had opted against fielding all four of their out-and-out fast bowlers, with James Pattinson the man who missed out on Friday.
Pat Cummins, however, proved expensive, with New Zealand opener Luke Ronchi severe on the paceman during a dashing 65.
Williamson showed touch play could be effective too when he swept off-spinner Travis Head for a four between two legside fielders.
Alongside Ross Taylor (46), he put on 99 for the third wicket.
Williamson hoisted John Hastings for a legside six to go into the 90s before a flicked four and single off the same bowler saw him to a 96-ball century, his ninth at this level, with eight fours and three sixes.
But he was then run out by Henriques's throw to Cummins.