Chasing 233 for the win, Australia fell to 44-5 before Green combined in a 158-point stand with Alex Careywho made 85, to relaunch their pursuit.
The all-rounder beat 10 fours and a six as Australia won the low-scoring contest with five overs to spare to go up 1-0 in the three-game Chappell–Hadlee Trophy series.
What a game 🔥From 44/5 at some point Australia fight back to clinch first ODI 👏Watch #NZvAUS on… https://t.co/9HUOYqV6yj
— ICC (@ICC) 1662466709000
Earlier, Glenn Maxwell claimed 4-52 to limit New Zealand to 232-9 after home captain Aaron Finch opted to play.
Maxwell took a superb one-handed hold at the backseat to fire Martin Guptill in the fifth before wreaking havoc with the ball.
Devon Conway (46), skipper Kane Williamson (45) and Tom Latham (43) missed their 50s but at 179-3 after 41 overs, New Zealand looked set for a late onslaught.
Maxwell, who fired Williamson just as the batsman came loose, fired Daryl Mitchell (26) and Latham at the same time and Josh Hazlewood grabbed three late order wickets to keep New Zealand under 250.
Australia found themselves in a hole early after the top half of their batting order was taken away in 12 overs.
Trent Boult (4-40) trapped Finch lbw for five, extending the fly-half drought that remains a major concern for Australia.
Boult then knocked out Steve Smith and Marnus Labuschagne in his successive overs before Matt Henry offered a double strike, which included David Warner’s wicket (20).
Green and Carey brought the hosts back into the contest with their positive stick, but more drama was on the cards.
Carey’s 99-ball blitz ended when he fired Lockie Ferguson halfway through, leaving Australia needing 31 runs from 65 balls with four wickets in hand.
Maxwell and Mitchell Starc went off in successive overs and Green developed cramps and was seen hobbling towards the end of his shot.
Green and Adam Zampa, who made 13, returned after a brief rain break and Australia triumphed when Ferguson, playing 46th, sent in a wide delivery.
The second game is scheduled for Thursday.
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