Australia were eventually restricted to 134 for nine as Corey Anderson took responsibility for the final over.
The result takes New Zealand to the top of their group. At one point, with Australia 44 without loss in the sixth over, the chance of victory looked as likely as spotting a yeti. The Black Caps have now reached base camp in their quest for the title.
To quote hometown boy, the Dalai Lama: "It is very rare or almost impossible that an event can be negative from all points of view."
Australia could take solace that they finished within cooey.
New Zealand's tactical nous, with Kane Williamson in charge for the first time at a world tournament, was impressive. On a smaller ground than Nagpur, where they defeated India by 47 runs, Williamson kept the spinners, himself, Mitchell Santner and Ish Sodhi out of the attack until the end of the six-over powerplay.
Yet the confidence in the spin saw them select bespoke opponents. The left-arm orthodox Santner and leg spinner Sodhi were used to take the ball away from the right-handers; Williamson brought himself on to bowl to left-handers Usman Khawaja and David Warner.
The bowling was impressive but arguably the key moment came when Khawaja, who had reached 38 off 27 balls was run out with an Adam Milne throw to Grant Elliott. Warner drove straight and looked for two but the stumps were successfully splayed. That gave the Black Caps the audacity of hope.
Seven balls later Warner picked out Martin Guptill at deep mid-wicket, leaving Australia 66 for four in the 11th over, having lost four wickets for 22 runs in 29 balls.
The most clinical dismissal saw Santner drift a ball past Steve Smith to have him stumped for six. That saw Australian coach Darren Lehmann sit up straighter than a yogi.
Santner finished with two for 30 from his allotment.
New Zealand posted 142 for eight after choosing to bat in the countries' first encounter in six World T20 tournaments.
The Kiwis raced to 58 without loss in their powerplay. However, Australia eased back into the contest with consistent bowling and flawless fielding, including the run outs of Elliott and Santner in the latter overs.
The opening partnership harvested 61 between Williamson and Guptill before they slumped to 76 for three, losing those wickets in 19 balls.
The batting highlight came with Guptill's dispatch of Agar for three sixes in the third over. While the deliveries were not wrapped in pastry, Guptill is unlikely to receive juicier pies in his career. He munched them as Agar was put in an 18-run oven.
A Guptill pull from James Faulkner deserved to be hit to the boundary but ended with Glenn Maxwell coasting across from long on to pouch a catch. Two slower balls from Maxwell accounted for Williamson (24 from 20) and Anderson (three from six) as Agar lined up steepling catches at long off.
Weather patterns can be fickle at the venue but it produced a perfect alpine day. Long sleeve sweaters were expected to be mandatory, but sunscreen appeared a wiser option.
Unfortunately, and perhaps as an indictment to tournament planning, swathes of fluoroscent seating was visible, despite the match being fought between the sides who contested last year's World Cup final.
Shane Watson's parsimony, with 12 dot balls in his one for 22, was the Australian bowling highlight alongside the wicket-taking of Maxwell and Faulkner. The latter two each finished with figures of two for 18 from three overs.
New Zealand opted to leave out Nathan McCullum and bring back McClenaghan in a crucial change from the side which beat India in their opening match.
Australia chose the spin trio of Adam Zampa, Agar and Maxwell.
Match centre with scorecard, wagon wheel and Manhattan/Worm
