Kyle Mayers top scored with 39, while Odean Smith chipped in 27 crucial late runs after the visitors started strongly but faltered mid-match in the face of quality bowling.
Mitchell Starc and Pat Cummins both took two wickets apiece.
After Australian skipper Aaron Finch won the toss and sent the West Indies in at Metricon Stadium, they smacked 14 off the opening Starc over in an aggressive start.
Hazlewood bowled Johnson Charles for three on his first ball, but Starc then dropped Mayers on 17 in the deep during the same over.
It proved costly as the stocky opener powered them to 47-1 off the first five overs, including a huge six off Cameron Green.
But when the dangerous Hazlewood returned to the attack, he struck again, removing Brandon King lbw for 12 to snap what was shaping up as a dangerous partnership with Mayers.
King should have reviewed the decision, with a replay showing the ball going over the stumps.
The tidy spin of Adam Zampa and Glenn Maxwell successfully put the brakes on before Cummins bowled dangerman Mayers for 39 off 36 balls as he swung wildly in a bid to keep the runs flowing.
They reached half-way at 68-3 and slipped further into trouble when captain Nicholas Pooran (2) was lbw to a Starc yorker on review after the umpire initially gave him not out.
With Australia on top, Raymon Reifer hit 19 but fell in the 15th over to Green, with Matthew Wade taking the catch, and Cummins accounted for Jason Holder who smashed two sixes in the seven balls he faced.
Hazlewood then snapped up Rovman Powell, caught and bowled, before Smith’s late flurry.
The hosts headed into the two-match contest on the back of a 2-1 series defeat to India last month, while the West Indies most recently lost by the same score to New Zealand in August.
Both teams are using the games as final preparations for the World Cup, which starts in Australia on October 16.
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