Hazlewood's first two overs went for 20, but he bounced back superbly to grab 6-52 from the nine he sent down as New Zealand collapsed from a commanding 4-254 - losing their last six wickets for 37 runs.
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.
The Black Caps were pressing hard for victory when the weather interrupted them, Australia having fallen 25 behind the Duckworth/Lewis par on 53 for three in nine overs.
Hazlewood ensured the opposition didn't see out their allotted overs, mopping up the tail with three wickets in four balls.
However, the skipper was furious at the performance of his bowlers, as the returning Mitchell Starc (0-52) and Pat Cummins (1-67) in particular, struggled with the line and length albeit on a ideal batting wicket.
Incredibly, Australia could not take advantage and when play resumed following a lengthy delay for rain, they were quickly in trouble with David Warner, Aaron Finch and Moises Henriques, all dismissed cheaply.
"I'd have preferred to be in New Zealand's position when we came off there", said Australia captain Steve Smith, who was 8 not out.
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While Australia are always a tough team to crack, the Kane Williamson-led side will head into the eight-team tournament with renewed confidence, following their six-wicket warm-up win over Sri Lanka at the same venue.
A rain delay of nearly two hours reduced the match to 46 overs per side but Ronchi looked as though he had never left the middle as he continued to fire once played resumed. "We perhaps got away with one there".
New Zealand were now 117 for two in the 16th over after a stand of 77 in 60 balls between Ronchi and Williamson.
But with the tournament looming as one for the batsmen, some tightening of the attack at the top of the innings will be necessary if coach Darren Lehmann wants his team to avoid a repeat of Australia's 2013 Champions Trophy campaign, which ended in the group stages.
"I think the first 10-15 overs, we bowled too full and both sides of the wicket and in England you've got to bowl pretty straight".
Starc didn't take a single wicket on his return to global cricket following the injury he picked up against India earlier in the year.
Latest news from the Champions Trophy Group A clash between Australia and New Zealand.