Starting the day on 25 for two, the West Indies built some stability into their innings thanks to Shane Dowrich (70) and Marlon Samuels (74) but there was little other resistance with wickets falling at regular intervals.
The Windies were all out for 216 in their second innings, giving them a lead of 46 following their 148 from the first innings, which Australia quickly polished off to claim a comfortable victory.
Darren Bravo’s dismissal for five saw the hosts rocking on 37 for three early in the day before Dowrich and Samuels steadied the ship.
When their partnership was broken the scoreboard was on 181 when Josh Hazelwood had Dowrich caught by Shane Watson.
Jermaine Blackwood followed shortly afterwards for 12, stumped by Brad Haddin off the bowling of Nathan Lyon with Samuels’ dismissal on the same score of 198 plunging West Indies back into trouble.
The last four wickets fell for just 18 more runs as Mitchell Starc polished off the tail to record the pick of the Australian bowling figures with four for 28.
Australia’s first-innings total of 318 had put them in control, and after restricting the hosts’ runscoring again on the third day, only had to hit a small target of 47 to win the match.
The visitors looked to be easing to victory – as they looked to wrap it up quickly with the threat of rain in the air – but they were unable to do so without losing a wicket as David Warner was caught at first slip by Bravo off the bowling of Jerome Taylor.
That brought Steve Smith to the crease, who got off the mark with his first ball thanks to a streaky four.
It also meant that Australia needed just one more run to win, and that came at the end of the over by Taylor, with Smith the man to wrap things up.
The second Test begins on June 11 in Kingston, Jamaica.