Australia are in Group A with England, Bangladesh and New Zealand, with the top two teams progressing to the semi-finals.
The Australians left Birmingham straight after the game for the bus trip to London where they will face Bangladesh in a must-win game on Monday at The Oval.
New Zealand had lost 7-37 in the final 5.5 overs, after Williamson was run out, and collapsed to what seemed a manageable total in the 46 overs Australia would have had, after the first rain delay reduced the match.
Earlier, New Zealand elected to bat under overcast conditions and got off to a brisk start.
Taylor went slogging for 46, and Williamson nudged a single for his ninth ODI hundred before departing moments later after deliberating with Neil Broom as they tried to nick a single. Josh Hazlewood picked up his career best 6 for 52, cleaning up the New Zealand tail.
Hazlewood took three wickets in four balls to wrap up the innings with an over to spare. There were three rain breaks during the course of play.
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Cummins, however, had already proved expensive by the time rain stopped play with New Zealand 67 for one midway through the 10th over. He played late, took chances when the odds were in his favour, manipulated the field and, when he tried to go big, he always had a well-constructed back-up safety shot. Starc's underarm throw and Matthew Wade's fumble saw them botch a certain run out with Ronchi stranded, then Starc dropped him at mid-on.
New Zealand have a couple of decisions to make.
The Black Caps didn't quite manage it, and they would have wanted more especially after Williamson and Taylor put on 99 for the third wicket.
His second fifty came off just 34 balls. Trent Boult survived the hat-trick but was then caught behind next ball.
Four years ago, when Australia and New Zealand game in the Champions Trophy 2013 was washed out, Australia had a slight advantage over their rivals and would not enjoyed the one point.
"He's a little bit slower than the others and hits the seam and has some good skills throughout the middle as well".
"I don't think we'll end up going with the four big quicks", Smith said.
"For us it´s trying to attack it as best we can".
"Those guys can come off, which they do majority of the time".