The hosts, who started Day 3 at 25 for 2, needing 146 more runs to erase the deficit and make Australia bat again, lost Darren Bravo early for 5. It was then when Marlon Samuels (74) partnered debutant Shane Dowrich (70) to put up 144 runs for the fourth wicket, the highest partnership in the match.
The West Indies threatened to take the match into the fourth day and put up a challenging total but Australia spoilt their plans in the post-tea session.
The hosts lost their last 6 wickets for just 18 runs, managing to extend the lead to just 46 runs. The visitors lost just one wicket to reach their target in 5 overs. David Warner was the only wicket to fall.
Mitchell Starc was the peak of the Australian bowlers with 4 for 28. Mitchell Johnson, Josh Hazelwood and Nathan Lyon took 2 wickets each.
The West Indies have not won a Test against Australia since 2003 while the last time they had drawn a series against Australia was in a home series in 1998. The Caribbeans' last series victory against Australia had come in 1993.
Australia's Adam Voges, who hit a century in his debut Test, was adjudged the man of the match.
The second and final Test of the series begins in Kingston on June 11.