But the remainder of Australia's top-order failed to stem the flow of dismissals, throwing their wickets away with reckless strokeplay and poor shot selection.
The Aussie men had plenty of time to knock off the final 50-odd runs, but seemingly opted to try and chase the target with boundaries, a move which ultimately backfired.
Finch described the batting performance as "very disappointing", suggesting the side's middle-order "panicked".
"Anytime you lose 6/19, you're going to struggle to win a T20 game," the Australian captain sheepishly conceded after the loss.
"We just needed somebody to take it upon themselves and get right through to the end.
"We needed under four-an-over for a majority of that last bit … just a bit of game-smarts would have just gotten us over the line.
"We've been working on trying to be a little bit more aggressive through the middle overs, but maybe we just have to rein it in."
West Indies spinner Hayden Walsh was a standout performer in the field, finishing his four-over spell with career-best figures of 3/23, including the dismissals of Ben McDermott, Dan Christian and Marsh.
Local speedster Obed McCoy then cleaned up the tail, claiming 4/26 in the second innings.
Former Australian spin bowler Brad Hogg tweeted: "Disappointing, game in the bag and thrown away. Finishing is the spots up for grabs. Opportunities are there and go to be taken."
Earlier in the contest, West Indies all-rounder Andre Russell slapped his maiden 50 in T20 internationals, cracking five sixes and three boundaries to guide the hosts to 6/145 in tough conditions.
Josh Hazlewood was superb in his return to the national T20 side, conceding just 12 runs from his four overs with precise line and length bowling.
The New South Wales seamer accounted for the wickets of opener Evin Lewis, T20 icon Chris Gayle and Russell.
"The start we got with the bat paved the way to just knock it around and play with a bit of cricket-smarts, which obviously we didn't do," Hazlewood said after the match.
"We take pride in our batting ... but unfortunately we couldn't get it done."
The second T20 will commence at 9.30am AEST on Sunday morning, with the West Indies taking an early 1-0 lead in the five-match series.