Marsh top-scored with 85 and was man of the match in Wednesday's day-nighter at Mount Maunganui.
New Zealand were 3-6 in the second over and in dire straits after losing the toss, with Ben Dwarshuis on a hat-trick.
The Australian captain stuck with his field-first policy, despite Tauranga's Bay Oval historically favouring the side that bats first.
That looked like it might hold as Tim Robinson led the home team's recovery. He scored his first white-ball international century, ending the innings with a four and a six to reach 106no from just 66 balls, with six fours and five sixes.
Robinson and Daryl Mitchell (34) had a partnership of 92 - a fourth-wicket record for Australia-New Zealand men's T20s - as the home side reached 6-181 in cold and blustery conditions.
The Australians' fielding was below-par, with several spilt catches.
While Travis Head dropped two chances, he was also involved in the highlight of the match late in the New Zealand innings.
Marcus Stoinis stopped a boundary and flicked the ball between his legs to Head, whose excellent throw caught Bevon Jacobs just short for 20.
The absence of Glenn Maxwell with a fractured arm could have been a significant blow to Australia as the run chase looked tricky.
But Marsh and Head (31) piled on 67 in the first six overs. Matt Short (29) then joined his captain in a 68-run partnership that ensured Australia's win.
A century went begging for Marsh when he holed out to Robinson at deep cover, his knock lasting 43 balls and featuring nine fours and five sixes.
Australia cruised to 4-185 from 15.3 overs, with the next two games in the Chappell-Hadlee series on Friday and Saturday at the same venue.
Dwarshuis took 2-40 and New Zealand's Matt Henry snared 2-43, while Australian leg-spinner Adam Zampa only gave up 0-27 from his four overs.
The tourists wore black armbands for Alex Carey, whose father Gordon died last week.
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