The Aussies had come into the tournament not expected to make the semi-finals, let alone lift the trophy, having lost successive T20 series 4-1 to Bangladesh and the West Indies, while the side has never mastered the shortest format.
Coach Justin Langer was also under pressure, with murmurings about team disharmony in the lead-up, while Australia were belted by England in the group stage too.
Finch had won the all-important toss, opting to bowl first, given 11 out of 12 sides chasing had won at Dubai this tournament.
The Black Caps started slow but skipper Kane Williamson accelerated their innings with 85 from 48 balls, plundering Mitchell Starc (60-0) all over the park, while Josh Hazlewood, who dropped Williamson on 21, contained them with 16-3 along with Adam Zampa (26-1).
Finch lost his wicket early, but David Warner (53 from 38) was excellent alongside Marsh, who hit a six off the first ball he faced, before Glenn Maxwell hit the winning runs.
“This is huge, to be the first Australia team to do it,” Finch said after the game. “So proud of how the guys went about the campaign.
“Backs were against the wall. We had to fight and certainly did that, had some great team and individual performances.”
Warner was named Player of the Tournament, despite pressure to lose his spot after a lean Indian Premier League and struggling in the warm-up games.
The Australian opener hit 289 runs including three half-centuries across the tournament, but Finch also reserved praise for Zampa, who finished with 13 wickets and an economy rate of 5.81.
“Can’t believe people wrote him (Warner) off a couple of weeks ago, it was almost like poking the bear,” Finch said. “(Zampa) Player of the tournament for mine, controlled the game, got big wickets, super player.”
Williamson offered credit to the Australians, admitting he felt their total would be competitive.
“We were trying to get a platform, the surface was holding a fraction,” he said. “Nice to build some partnerships and get what we thought was a competitive total, only to be chased superbly. They’re a fantastic side, had a brilliant campaign and came out and turned it on.”