After a hotly contested first two days, Australia dominated on Friday (Saturday morning AEDT) with Hazlewood taking 5-43 as the tourists bowled their hosts out for 141 in their second dig.
The Windies were thwarted by a collapse of 6-26 in pursuit of 301 for victory, but were boosted by Australia's frequent tormentor Shamar Joseph who cracked 44 runs off 22 balls - including four sixes - in a furious cameo at No.10.
Veteran spinner Nathan Lyon (2-20) struck twice in the final over of the day to seal a win that helped Australia overcome their shock loss in the WTC final and start the three-match series on a high.
The tourists are likely to regain the injured Steve Smith for the second Test that begins in Grenada next week after his absence was glaring on a tough batting deck earlier in the match.
But Australia posted a competitive second-innings total of 310 on day three thanks largely to middle-order trio Travis Head (61), Beau Webster (63) and Alex Carey (65).
In the final session, Hazlewood took all four scalps in a West Indies batting implosion of 4-9 that devastated the hosts' middle order and had the veteran on a hat-trick at one stage.
Hazlewood broke the best partnership among the recognised batters, a 43-run stand between Keacy Carty and John Campbell, when Campbell (23) played a daring lap shot straight into Carey's gloves.
On Hazlewood's next ball, debutant Brandon King (0) caught a high inside edge to Cameron Green, who made up for dropping a sitter by leaping high at gully.
Captain Roston Chase (2) could not replicate a solid first innings, caught from by Sam Konstas at short leg, before Hazlewood bowled the established batter Carty with a ball that seamed in on the tricky pitch.
Axed from the XI, Marnus Labuschagne made a valuable contribution as a substitute fielder, whipping a direct hit at the stumps from the other end of the wicket to run Alzarri Joseph (0) out.
Hazlewood completed his 13th Test "five-fer" when Jomel Warrican (3) edged to Carey the ball after surviving an lbw review.
The end appeared nigh for West Indies when unlikely top-scorer Joseph edged Lyon to Usman Khawaja in the slips.
Earlier, No.5 Head (61 off 95 balls) ignited Australia's fightback after the tourists went to stumps on day two in a precarious position at 4-92.
Head was trapped lbw by a low delivery from cult hero paceman Joseph (5-87), who was again the West Indies' most important bowler after his match-winning heroics in Brisbane 17 months ago.
Showing impressive patience, the usually furious Head had been given a life earlier, spilt by Justin Greaves on 21 as one one of the hosts' seven dropped catches for the match.
After bringing up a third Test half-century, towering allrounder Webster (63 off 120) fell to an unfortunate tickle down the leg side as Joseph picked up his third wicket.
Carey, displaying some exciting stroke-making, was out caught in the deep by Greaves off Chase - the only Australian wicket that fell to spin for the match.
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