Stoinis, who also took three wickets in New Zealand's 286 for nine from 50 overs, guided Australia from 54-5 in the 13th over to within sight of victory as he grabbed the game by the scruff of the neck.
The 27-year-old, playing his second one-day international, shared in an 81-run seventh-wicket partnership with James Faulkner (25) and 48 with Pat Cummins (36) for the eighth before Stoinis exploded for 31 runs in 11 balls to put his side into a position where they needed 19 runs from the final 24 balls.
Such was his dominance he and last man Josh Hazlewood shared a 54-run partnership, without the tall fast bowler facing a single delivery.
Hazlewood, however, was run out while backing up too far by New Zealand captain Kane Williamson on the final ball of the 47th over to dismiss the visitors for 280, giving New Zealand a six-run victory.
Stoinis was left stranded on 146 not out, having hit nine boundaries and 11 sixes.
Australia were already missing David Warner (rested), Usman Khawaja (in Dubai preparing for the tour of India) and Steve Smith (injured) but then lost stand-in captain Matthew Wade before play began with a back injury.
New Zealand had been well in control of the match as make-shift wicketkeeper Tom Latham repaid the faith of the selectors with three catches and two stumpings.
Latham, who was a wicketkeeper earlier in his career, took the gloves ahead of the uncapped Tom Blundell as the selectors looked to build more flexibility in their squad ahead of the Champions Trophy in England later this year.
The hosts had been humbled in a 3-0 series loss in Australia before Christmas but showed a marked improvement in their fielding and bowling as they put the visitors under pressure from the start of their run chase.
New Zealand's innings was anchored by Neil Broom's 73, while Martin Guptill added 61 at the top of the order.
The three-match series continues with further matches in Napier on Feb. 2 and Hamilton on Feb. 5.
(Reporting by Greg Stutchbury; Editing by John O'Brien)
