The tourists set their opponents a total of 145-9 on the Gold Coast and Australia looked a long way from reaching that when languishing on 58-5 in just the eighth over.
But Finch (58) and Matthew Wade (39 not out) put Australia back on top and they clinched a three-wicket victory after the latter was dropped twice in the final over.
Kyle Mayers earlier top-scored for West Indies with 39, while Odean Smith chipped in at the end with 27 from 17 balls, but a lack of runs in the middle overs ultimately cost them.
Josh Hazlewood (3-35) ensured Australia had a reachable target to aim for, but David Warner and Cameron Green went for 14 either side of Mitch Marsh (3).
That slow start continued until Finch and Wade combined for a 69-run stand. Finch was caught by Jason Holder, leaving Australia requiring 19 from the final 2.5 overs.
Losing Pat Cummins (4) piled the pressure on the hosts, but Wade was twice let off and, with two runs needed from the final two balls, managed to get his side over the line.
Finch closing in on milestone
Home skipper Finch has struggled with the bat of late and took on an unfamiliar role in the middle order as Australia experimented ahead of the upcoming T20 World Cup.
He did his confidence no harm as he was the only batter to pass 50 and is now just 27 runs away from becoming the first Australian to score 3,000 runs in men’s T20I history.
Hazlewood halts West Indies
Hazlewood claimed three wickets to become the third Australia player to take 50 in the format after Adam Zampa and Mitchell Starc.
That set the hosts on their way to a third victory in four T20Is against the Windies, with this making it back-to-back wins for the first time since March 2012.
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