Australia was set 192 to win in the final two sessions and got a solid start, but late wickets produced some nervous moments before a scampered single took them to 192-7 and earned a 1-0 lead in the three-match series.
"I think full credit has to go to our bowlers, not only the way they batted in our first innings but the way they came out and bowled in the West Indies second innings as well," Australian skipper Michael Clarke said. "No doubt that went a long way towards us winning the game."
Watson struck four fours and a six off 57 balls while Hussey hit two fours and two sixes. Ed Cowan laid the anchor with 34 off 100 deliveries.
Off-spinner Narsingh Deonarine gave the West Indies hope with 4-53 while fast bowler Kemar Roach took 2-45.
Earlier, the West Indies was bowled out for 148 on the stroke of lunch after resuming on 71-5. Roach provided the main resistance with a stubborn 25.
Ben Hilfenhaus, who had started the West Indies' slide on day four, ended with 4-27 while fellow pacer Ryan Harris took 3-31.
Australia's target was 192 off 63 overs but once heavy rain delayed the restart after lunch, it was always going to be a fight against time and light, rather than overs.
The visitors enjoyed some luck as they attempted to complete a come-from-behind victory. David Warner had two inside edged boundaries just past the stumps off the luckless Roach as he and Ed Cowan added 31 for the first wicket.
Darren Sammy eventually removed Warner to a wicket-keeper's catch, his thin edge neatly pouched by Carlton Baugh. Roach had two very close lbw appeals against Watson turned down, both by umpire Tony Hill and the Decision Review System, but the West Indies bowlers limited Australia to 61-1 at tea.
Watson also survived a sharp gully catch to Sammy off Roach when 5.
After the break, Australia showed increased urgency, with Watson dominating a second wicket stand of 75 with Cowan. Deonarine brought the hosts back into it with four top-order wickets in quick time.
Watson top-edged to deep square leg at 106-2 and Cowan, Ponting and captain Michael Clarke soon followed. Cowan pulled a catch to midwicket, Ponting (14) was bowled off the under-edge by one that kept low and Clarke (6) chipped a return catch.
At 140-5, the match was in the balance even as the sun started to dip in the west.
But Hussey provided much needed momentum with a busy innings, including two early reverse sweeps for boundaries. He and fellow left-hander Matthew Wade (18) settled Australian nerves with a stand of 37 for the sixth wicket that brought their team within 15 of victory.
But there was late drama as Roach returned to claim Wade, who slashed to deep point, and Hussey, whose stumps were rattled.
But Harris and Hilfenhaus, two architects of the turnaround on Tuesday, took Australia to victory.
The winning run was as dramatic as the previous four days, the umpires needing a TV replay to confirm that Hilfenhaus had managed to scamper home before a run out was effected.
Earlier in the day, Harris made a crucial strike inside the first 10 minutes when the West Indies resumed on 71-5. He trapped left-hander Deonarine plumb lbw for 21.
Baugh and Sammy shared a partnership of 31 to take the West Indies past 100 and lift home team hopes. But Sammy, dropped badly by Nathan Lyon at deepish mid-on off Siddle when 12, failed to make the most of his second chance and the skipper was bowled off the inside edge by Watson at 106-7.
Hilfenhaus, who had blasted out the top order on Tuesday, further dented the hosts when he claimed Baugh to a chipped catch to Harris at mid-on at 116-8.
Baugh struck two fours in 23 off 48 balls in an hour and a half. Roach battled hard to resist the accurate Australians and scored 25 in 83 minutes.
Fidel Edwards failed to show the same application, holing out to mid-off off Siddle after scoring just 3. Roach and Bishoo added 23 for the last wicket before Harris returned to bowl Roach off the inside edge.
Sammy said the test was decided by his team's meltdown on day four, when it allowed Australia's last three wickets to add 156 runs and the top order was blown away.
"I think we dominated for the first three days. One bad hour set us back behind the eight ball," Sammy said. "It's one we thought we had but we have two more and we have to come back and fight and look to be stronger."
The second test begins at the Queen's Park Oval in Trinidad on Sunday.