The South African-born left-armer, a fiery short-ball specialist, claimed seven for 39 as the West Indies crumbled to be all out for 134 in a first innings that lasted a mere 45.4 overs.
In reply, New Zealand were 85 for two at the close of play with Jeet Raval on 29 and Ross Taylor on 12.
Tom Latham (37) and Kane Williamson (one) fell in quick succession before Raval and Taylor cautiously added 17 in 8.3 overs through to stumps.
After the tourists safely progressed to 59 without loss, Wagner engineered a collapse that saw all 10 wickets fall in 22 overs for only 75 runs.
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He was twice on a hat-trick as he posted his first seven-wicket haul and bettered his previous best of six for 41 against Zimbabwe last year.
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The West Indies were full of confidence going into the match on the back of a recent series win over Zimbabwe and a Test victory against England, and the confidence grew as openers Kieran Powell and Kraigg Brathwaite progressed steadily passed the half-century mark.
By digging the ball in short and angling it towards the rib cage he had his first success in his third over when Brathwaite went for 24.
Two overs before lunch, when the tourists looked to be recovering at 75 for one, Boult took a leaf out of Wagner's book, using a short-ball to account for the West Indies top scorer Powell who made 42.
Wagner then hit a purple patch as he ripped through the West Indies' new generation of specialist batsmen in the space of four balls.
Roston Chase and Shane Dowrich mounted a short-lived revival with a 17-run partnership before Wagner was back in the action again with a rising delivery to Chase (five) that was steered to Jeet Raval at leg slip.
With Wagner's first five wickets all from short deliveries, new batsman Holder stood tall for the first ball he faced, only for Wagner to switch to a yorker which bowled him, and 59 without loss had become 97 for seven.
Kemar Roach (not out 14) put on 29 for the last wicket with Shannon Gabriel (10) before Wagner had Gabriel caught in slips to wrap up the innings.