Having won the toss and opted to bat, Martin Guptill got New Zealand off to a flying start and once he was dismissed by Hazlewood for 26, Luke Ronchi, having played a supporting role to that point, thrust himself to the fore.
Williamson's 100 off 97 balls anchored a New Zealand innings that promised a lot more, but Australia found their groove in the closing overs to take the final seven wickets for 37 runs.
Ross Taylor of New Zealand plays to the leg side as wicketkeeper Matthew Wade looks on during the ICC Champions Trophymatch between Australia and New Zealand at Edgbaston. "We had work to do and a quality bowling attack against us, so we perhaps got away with one there".
Paceman Josh Hazlewood was the pick of Australia's bowlers with 6/52 from nine overs.
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 global sides with an eight-wicket win over Bangladesh at the Oval on Thursday.
The batsman cashed in on the opportunity and struck Cummins for another four and a six in him next two consecutive balls to carry forward the momentum. "We need to keep that going and move on".
However, it was a morale booster for the Black Caps, whereas, the exact opposite for the Australian team. Hazlewood ended with six wickets to his name, ensuring his side snatched back the initiative from New Zealand's grasp with the wickets of Broom, Neesham, Milne, Santner and Trent Boult to add to the early wicket of Guptill.
"The last 10-15 overs were pretty good and that's something that I guess we have done pretty well for a very long time".
Egypt to suspend air links with Qatar from tomorrow
Qatar, however, claims that hackers took over the media outlet's website and published the comments, which it says were fake. Saudi Arabia is also shutting its border to neighboring Qatar, locking the state out of its only over-land border crossing.
New Zealand could not quite capitalise on the start as the innings lost its steam with the dismissal of Williamson.
Earlier, a century from captain Kane Williamson helped New Zealand to 291 before they were bowled out.
"Let's hope it's rust and let's hope it's gone", Smith said.
The former Australia A player rode his luck at times, including being dropped by Mitchell Starc at mid-on, before finally falling to John Hastings for a brisk 65 off 43 balls, with nine fours and three sixes, with Maxwell taking another catch.
If New Zealand gets their lineup right, they could win the Champions Trophy, 17 years after their one and only major success.
While batting first, Bangladesh scored 305 runs for 5 in 50 overs with Tamim Iqbal's 128 runs innings, recording his second Century against England and overall ninth.
Milne then removed Finch and Henriques, who both looked out of sorts.