It was a much better batting performance from the West Indies in their second innings, bowled out for 317, their best total of the series.
The loss means the West Indies lost the two-match series 2-0.
Resuming on 244 for 6, still needing 85 to make New Zealand bat again, West Indies lost their first wicket of the day in the second over with the score of 252.
Captain Jason Holder was bowled by Tim Southee for 61, playing down the wrong line to a delivery that hit the top of off stump. He had added only run one to his overnight total.
It was Southee’s first wicket of the second innings after taking 5 for 32 in the first when the West Indies were bundled out for 131.
With the score at 282, Southee picked up his second wicket when he had Alzarri Joseph caught down the legside by BJ Watling after a well-played 24 that came off 12 balls and included three fours and two majestic sixes square of the wicket.
The West Indies still trailed by 47.
Meanwhile, at the other end, Da Silva, who was making his Test debut,was a picture of composure. On 25 overnight, he moved onto his first Test half century with aplomb farming the bowling to protect Chemar Holder, who was also playing in his first Test.
The 50th run was long in coming. After almost two overs on 49, he dropped a Trent Boult delivery into the onside for a single. His maiden half century came up in 77 balls and included five fours.
He celebrated his milestone by smashing a Daryl Mitchell delivery straight down the ground for his sixth four which also brought up 300 runs for the West Indies.
He was eventually trapped lbw by Neil Wagner for 57 to leave the West Indies on the verge of defeat at 307 for 9. He faced 84 balls in his 135-mimute stay at the crease.
Chemar Holder, who had edged a Southee delivery to get off the mark and then brutally hit Mitchell down the down the ground for consecutive fours, remained unbeaten on 13 after Wagner bowled Gabriel for 0 to seal a comprehensive victory.
Wagner finished with 3 for 43 while Boult took 3 for 97.
Scores: New Zealand 460 (Henry Nicholls 174, Neil Wagner 66. Gabriel 3 for 93). West Indies 131 ( Blackwood 69; Southee 5 for 32, Jamieson 5 for 34) and 317 (Campbell 68, J.Holder 61, Da Silva 57; Wagner 3 for 54, Boult 3 for 96).
When bad light stopped play on Saturday with just over 17 overs left in the day's play, Holder was unbeaten on 60 and debutant Joshua Da Silva on 25 as the West Indies closed the day on 244 for 6, still trailing New Zealand by 85 runs with only four tail-end wickets in hand.
Earlier, John Campbell made his highest Test score of 68 as the West Indies showed improvement on their embarrassing first innings display.
Resuming on their overnight score of 124 for 8 in reply to New Zealand’s 460, the West Indies added only seven runs to be all out for 131.
Tim Southee had DaSilva caught behind on his overnight score of 3 and bowled Shannon Gabriel for 2 to finish with 5 for 32. Chemar Holder remained unbeaten on 8. Kyle Jamieson, who did most of the damage on Friday, took 5 for 34.
Following on, the West Indies had a much better -looking start getting to 37 when Trent Boult took two wickets in an over to reduce the visitors to 41 for 2. Kraigg Braithwaite had got to 24 when he flicked the New Zealand quick off his legs only to be caught low down at leg gully by a diving Will Young.
Three balls later, Darren Bravo, on four, fended off a short delivery to point where Henry Nicholls took a dolly.
But just as it seemed as if the West Indies were in for another swift capitulation, Campbell and Shamarh Brooks mounted an 89-run stand that took the West Indies to 130 when a length ball from Neil Wagner slanted across Brooks, who went forward to defend only to see the ball take the edge and carry through to wicketkeeper BJ Watling. He made 36.
Roston Chase came and faced seven balls before edging Jamieson to Tom Latham at slip without scoring. Jamieson then bowled bowled Campbell off the inside edge for 68 as the West Indies slipped from 130 for 2 to 134 for 5.
Jermaine Blackwood, whose 69 made up more half the West Indies first innings score, threatened once again to take the attack to the New Zealand bowlers but after racing to 20, he was bowled by Boult trying to slog a ball that swung and smashed into his stumps. The score was then 170 for 6 and the West Indies were sliding towards defeat inside three days when Holder had his best showing of the series.
He and Da Silva have so far added 74 for the seventh wicket and will resume on Sunday hoping to add many much-needed runs to their face-saving stand and perhaps pray for rain.
Boult has so far taken 3 for 75 while Jamieson has 2 for 43 and Wagner 1 for 53.
Sompal Kami knew that two runs from the final ball of the match would secure an unlikely triumph, but batting partner Ghulsan Jha was run out at the non-striker's end to conclude their first meeting in international cricket.
South Africa were restricted to 115-7 with opener Reeza Hendricks producing 43 runs from 49 balls, ultimately handing the initiative to their opponents to keep their Super 8s hopes alive.
Nepal would last until the eighth over before losing Kushal Bhurtel and Rohit Paudel in three balls, but the Rhinos would steady the ship, leading to a nail-biting finish.
However, Tabraiz Shamsi (4-19) struck twice in a game-changing 18th over, aided by Ottniel Baartman denying seven runs in the final over to decide the tie.
South Africa face the United States in Antigua on June 19, while Nepal search for their first win of the tournament as they take on Bangladesh in their final Group D game.
Meanwhile, New Zealand secured their first win of their T20 World Cup campaign with relative ease, beating Uganda by nine wickets at the Brian Lara Cricket Academy.
The Black Caps exit from the tournament had already been confirmed following Afghanistan's seven-wicket triumph over Papua New Guinea, exiting a World Cup before the last four for the first time since 2014.
Trent Boult dictated the early pace of the encounter, taking two wickets inside the first over as their opponents quickly saw five wickets fall for just 15 runs.
Kenneth Waiswa was the only batter for the Cricket Cranes to reach double figures, as wickets from Tim Southee (3-4) and Mitchell Santer (2-8) in successive balls saw Uganda all out for 40.
Kane Williamson's side faced just 33 balls on their way to victory, as Devon Conway scored an unbeaten 22 from 15 balls to win the contest ahead of their final Group C fixture against the Barramundis.
The Phoenix restricted the Rockets to 118 all out off their 100 balls after winning the toss and putting them in to bat.
Southee, along with fellow pacers Adam Milne and Chris Wood, started brilliantly for the Phoenix as Tom Banton (5), Alex Hales (9), Joe Root (2) and Rovman Powell (1) all fell within the first 30 balls to leave the Rockets 22-4.
Tom Alsop and Imad Wasim steadied the ship with a 77-run fifth wicket partnership but the innings came to an end quickly after the wicket of Alsop and Wasim being retired out.
Southee was brilliant on his way to figures of 5-12 from his 20 deliveries while Wood took 2-30 off 20 balls.
The Phoenix then needed 93 balls to reach 120-4. They got a good start from England opener Ben Duckett who made 30 off 16 balls before he fell to Luke Wood 17 balls into the reply.
The wickets of Jamie Smith (0), Moeen Ali (13) and Dan Mousley (0) followed soon after bringing Liam Livingstone and Jacob Bethell to the crease together.
The pair then saw their team home, with Bethell finishing 38* off 29 balls and Livingston finishing 30* off 32 balls.
Luke Wood and John Turner took two wickets, each for the Rockets.
With one game left before the knockout rounds, the Phoenix are third with 10 points, just one ahead of the Northern Superchargers, while the Rockets are fifth with six points.