Australia, who won the toss and elected to bat, scored 189/6 in 20 overs. Finch and Marsh shared a 114-run partnership for the second wicket to lead Australia to their strong total.
West Indies took off quickly towards the target, reaching 75/1 in the first six overs of power-play with openers Evin Lewis (31 off 14) and Lendl Simmons (72 off 48) hammering Aussie bowlers.
However, the visiting team bounced back by picking wickets regularly. At 132/5, West Indies faced an uphill task, needing 68 off 27 deliveries.
Andre Russell (24 off 13 balls) and Fabian Allen (29 off 14) kept Windies in the hunt adding 47 for the sixth wicket in 3.3 overs.
The 26-year-old Allen hit three sixes while Russell hit one off Riley Meredith's fourth over -- the innings' 19th -- to take 25 runs and leave the home side with just 11 to get in the final over bowled by Starc.
With Allen out on the last ball of Meredith's over and bowler Hayden Walsh in at non-striker's end for the final over, Russell refused to take singles and instead tried to hit big sixes. But Starc's low full tosses were difficult to get away and Russell managed a six only on the last delivery.
Brief scores Australia 189/6 in 20 overs (M Marsh 75, A Finch 53, H Walsh 3/27) beat West Indies 185/6 in 20 overs (L Simmons 72, E Lewis 31, F Allen 29, A Russell 24, M Marsh 3/24, A Zampa 2/20) by four runs.
Gros Islet (St Lucia), July 15
Mitchell Marsh's 75 off 44 balls (4x4s, 6x6s) and Aaron Finch's 53 off 37 deliveries (5x4s, 3x6s) helped Australia register a four-run win over West Indies in the fourth T20 International here late on Wednesday night.
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