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Finch returned to form with a punishing knock of 69, and Maxwell blasted 70 from 31 deliveries.
Maxwell’s mad innings was full of shots not described in any textbook, and it helped the visiting side post a formidable tally of 208 for four.
New Zealand briefly clung to hope while Martin Guptill (43) and Devon Conway (38) were weaving their magic.
But the chase petered out when spinner Ashton Agar nabbed three wickets in the 13th over.
The left-armer finished with career-best figures of six for 30 as New Zealand capitulated to be all out for 144.
It was only the fifth time in the history of international T20 someone had taken six wickets.
Paceman Riley Meredith played a key role as well.
He took a wicket in his opening over of international cricket, and removed Kane Williamson in his second.
The 24-year-old right-armer was getting the ball through at 150kmph and his two for 24 helped make a difference.
But Finch and Maxwell set up the win.
Australia’s captain needed some early luck, though.
Finch survived a review for lbw in the opening over from Tim Southee.
The ball was hitting but not by enough.
Umpire Chris Gaffaney had called not out and that is how it stayed.
Trent Boult helped make up for the disappointment in the next over.
He drew Matthew Wade into a drive and nicked him out with the outswinger.
The Black Caps were on top and the ball was swinging.
But Josh Philippe was swinging as well. Finch got a shot out of the middle as well and suddenly all his troubles faded away.
With a previous high score of 18 this year, the accomplished right-hander was due and dispatched a series of shots down the ground.
The power play was perhaps a little underwhelming all the same.
But with left-arm spinner Mitchell Santner ruled out with a head cold, the Black Caps leaned on all-rounder Jimmy Neesham and he got punished.
And Kyle Jamieson is a howitzer in test cricket but a Tom Thumb in T20s.
It all added up to a very painful exit from the power play for the home side.
Philippe had taunted the field with several skyward shots which evaded a set of hands. But on 43 Guptill got under one.
Ish Sodhi made the breakthrough but Finch was loosening his shoulders. He brought up his 50 with a superb switch hit for six and then dispatched Boult for three boundaries in the 14th over.
Maxwell unleashed his creativity at the other end and, with five overs remaining, Australia had a massive total in sight.
Finch’s departure did not help any. Perhaps it just made Maxwell angry because he clubbed Neesham for 28 in an over — that’s 4,6,4,4,4,6. Neesham’s four overs ended up going for 60 — the second most expensive for a New Zealander.
Maxwell finally miscued but it was special innings. Five sixes, eight fours and just 31 deliveries. That is what you call carnage.
New Zealand leads the series 2-1. Game four is in Wellington tomorrow.