The tourists, who resumed the day at 158-4, lasted only 35 deliveries before being all out for 193 with left-armer Boult producing career-best figures of six for 40.
His last four wickets came in the space of six fast, inswinging deliveries that left the West Indies 248 runs short of New Zealand's 441 and captain Brendon McCullum did not hesitate to enforce the follow on.
When a brief shower forced an early lunch on the third day, the West Indies were 45 without loss with Kieran Powell on 23 and Kirk Edwards not out 17.
But, with the first delivery of his fourth over, Boult found the outside edge of Deonarine's bat, Ross Taylor took the catch at first slip and the collapse was underway.
Boult could have had the eventual not out batsman Denesh Ramdin with the next ball when a leg before wicket appeal was turned down. New Zealand decided not to seek a referral but replays showed the batsman would have been gone.
Darren Sammy was bowled on the second ball he faced and Shane Shillingford followed immediately. Tino Best survived the hat-trick ball but was bowled by another swinging Boult delivery on the second ball he faced.
Boult, a 24-year-old left-arm seamer, has taken 63 wickets at 28.00 in his 19-Test career, with his previous best figures being 6-68 against England earlier this year.
The West Indies were also forced to follow on in the first Test in Dunedin last week and fought back in their second innings to salvage a draw.
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