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Marsh scored 81 from 97 balls on his debut to earn the man-of-the-match award, and Brad Haddin hit an enterprising 50 off 52 balls, as Australia reached 273 for eight from their allocation of 50 overs after they were sent in to bat.
Gaining modest bounce and little sideways movement from the hard, easy-paced Arnos Vale Sports Complex pitch under cloudy skies, Australia, led by Nathan Bracken, bowled steadily to dismiss West Indies for 189 from 39.5 overs.
The victory gave Australia a 1-0 lead in the five-match series which continues on Friday at the Queen's Park Stadium in the Grenada capital of St. George's.
But the near capacity crowd became incensed, when Bracken bowled Darren Sammy behind his back for 33.
Sammy needed confirmation from umpire Asad Rauf as to the method of his dismissal, since he did not know whether the ball came off wicketkeeper Brad Haddin's pads, or had hit the stumps directly.
Play was delayed for close to 15 minutes before police and army officers moved in and regained control of the situation.
Australia made sure there was no comeback story for a West Indies batting line-up without Shivnarine Chanderpaul, after they restricted them to 29 for three in the ninth over.
Brett Lee delivered two vital blows, when he had Xavier Marshall caught behind for six in the third over with a delivery that bounced and moved away from the opener - a similar delivery which got rid of Ramnaresh Sarwan.
In between, Bracken had Chris Gayle lbw for 20 with a delivery that moved back from outside the off-stump.
Australia continued to apply the pressure, but Dwayne Bravo and Andre Fletcher added 51 for the fourth wicket to give the West Indies innings some backbone before three wickets fell for 31 runs in the space of 48 balls set the home team back further.
Australia were then put on the backfoot, when Sammy joined Denesh Ramdin to add 52 at a run-a-ball for the seventh wicket before the bottom of the boat fell out.
Bracken was the most successful Australia bowler with four wickets for 31 runs off 5.5 overs, and Lee captured two for 37 from nine overs.
Marsh had given Australia early momentum, when he shared 75 for the first wicket with Shane Watson, but the visitors' innings appeared to be at the crossroads, when they declined to 140 for four in the 31st over.
But a fifth-wicket stand of 91 between Haddin and Mike Hussey, who made 44, put the Aussies back on track.
Marsh drove Dwayne Bravo through cover for the sixth of his seventh boundaries to reach his 50 in the 15th over, and he become the eighth Australian to score a half-century on debut in an ODI.
Australia then entered a period where West Indies starved them for boundaries. They endured almost 15 overs without either a four or a six, but gained traction with nimble running between the wickets.
But Haddin entered and struck four fours and a six to give a boost to the innings before he was one of four wickets that tumbled in the closing overs as the visitors chased quick runs.
Darren Sammy was the most successful West Indies bowler with two wickets for 31 runs from seven balls.
Australia's batsmen tried to maintain the frenetic scoring rate set by their openers, but West Indies claimed three wickets to get back into the match.
Ricky Ponting was bowled for six, when he dragged a ball from Jerome Taylor into his stumps in the 19th over.
Michael Clarke chased a wide ball from Sammy and was caught behind for nine in the 25th over.
And Marsh was caught at cover off Sammy, when he drove a slower ball straight into the lap of West Indies captain Chris Gayle in the 31st over.
Haddin came to the wicket and turned things around for the Aussies with some enterprising batting.
But he was caught at long-off off Sulieman Benn in the 45th over, and Hussey fell in similar fashion in the 48th over for 44 from 58 balls to expose the Australian tail.
Cameron White was lbw to Sarwan for eight in the 49th over, and James Hopes was bowled by Bravo for nine off the third-last ball of the innings in the late flourish.