The Black Caps owed a lot to the bowling of Trent Bolult who took 10 wickets in the match after claiming six in the West Indies first innings which could only reach 193 in reply to 441.
Boult took four more in the second innings as the tourists were once again bowled out cheaply, this time falling for 175.
With one Test remaining and after the opener ended in a draw in Dunedin victory in Wellington put New Zealand in a strong position to take the series.
Boult and Tim Southee were the chief destroyers on the third day.
The tourists had resumed in the morning on 158 for four but failed to add significantly to their total.
Boult started the rot when he removed Narsingh Deonarrine for 22 with the total on 175.
West Indies went on to lose their lost five wickets for the addition of just 18 runs.
Boult took the wickets of Darren Sammy, Shane Shilingford and Tino Best, all for ducks, before Southee removed Shannon Gabriel who also failed to trouble the scorers.
That left the tourists on 193 all out and when they were asked to follow on things looked a little better by the lunch interval.
They reached the interval on 45 without loss thanks to openers Kirk Edwards and Kieran Powell.
But when Powell was bowled by Southee for 36, with the total on 74, a familiar pattern had started to form.
Southee also removed Edwards and Marlon Samuels, with the other wickets to fall before tea going to Boult and Neil Wagner, West Indies having struggled to 135 for five which was still 113 runs short of making New Zealand bat again.
The tourists were left looking to the experience of Shivnarine Chanderpaul and Denesh Ramdin if they were to have any chance of making New Zealand bat again.
Chanderpaul proved to be the pick of the West Indies batsmen as he finished the innings on 31 not out and a forceful 21 late on from Tino Best helped to add a small modicum of respectability to the total.
But neither could prevent the home side from taking the victory and going 1-0 up in the series with the third Test due to begin in Hamilton on December 19.