Rain came to Australia's rescue as the match got washed out when they were struggling at 53 for three in nine overs, chasing a revised target of 235 in 33 overs.
"I thought it was probably one of the worst bowling displays that we've put on for a very long time," a disappointed Smith said after the match.
"We bowled both sides of wicket. We gave them a lot of freebies. And it was pretty ordinary, to be honest with you."
The Kiwis were 254 for three in the 40th over before Williamson's run out sparked a slump and they could post 291 all out.
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This was Mitchell Starc's first match since the second Test against India in March and Smith hoped that he would come good in their second ODI against Bangladesh.
"Let's hope it's rust and let's hope it's gone. Because that was pretty ordinary. Credit to them they came out quite hard and played quite well up top and got their innings going.
Smith admitted that rain came to his team's rescue.
"We still had a lot of work to do and they've got a quality bowling attack as well, so we perhaps got away with one there. We certainly would have preferred to be in New Zealand's position when we came off at the end," he said.
Australia and New Zealand have earned a point each after the washout.
"It's obviously not ideal for both sides to have a washout. For us now it's just about making sure that we're playing each game like a final and winning the next two.
"The guys were going well with the ball in hand and it would have been interesting to see how it unfolded but we can't control the weather.
"We thought today was pretty good, and it's important we move on to the next game. The nature of this tournament is every game is basically knockout-style. We'll need to be better again going into the game against England," said Smith.