The beanpole six-foot, seven-inch left-arm spinner claimed five for 93 as the Black Caps, winning the toss and batting first, were dismissed for 293, 14 minutes before the scheduled close at Kensington Oval, reports CMC.
Left with seven overs to negotiate, West Indies reached the safety of the close of play without loss, openers Chris Gayle (18) and rookie Kraigg Brathwaite (11) remaining together.
Using the Oval's 50th Test match as his stage, Barbadian Benn chose an ideal time to produce his best performance of the series, which is on a knife's edge at 1-1.
Extracting bounce and turn in an extended spell in the post-lunch session, Benn ripped through the New Zealand innings to send the tourists sliding from 95 for two at lunch to 194 for seven at tea, as they lost five wickets for 99 runs.
All-rounder Jimmy Neesham top scored with 78 to help prop up the lower order while tail-ender Mark Craig finished unbeaten on 46.
Ross Taylor, with 45, and Kane Williamson, 43, both came within sight of half-centuries before falling while captain Brendon McCullum got a start with 31 but failed to carry on.
Fast bowler Kemar Roach, though erratic, finished with four for 61.
It was Roach who made the early breakthroughs after the Black Caps chose to bat first on another flat surface, removing both openers cheaply inside the first hour.
He had Hamish Rutherford caught at mid-on for four, lazily driving to Shiv Chanderpaul, in the fourth over of the morning with the score on 17.
Left-hander Tom Latham followed soon afterward in Roach's fifth over for 14, lbw to one that hurried onto him as he got neither back nor forward.
Two overs earlier, the batsman was handed a reprieve thanks to the umpire's Decision Review System (DRS). He was given out lbw to Roach but replays showed the ball pitched out outside leg.
Taylor and Williamson then put on 74 for the third wicket, blunting the attack and taking New Zealand to lunch at 95 for two.
Williamson, especially, grew in confidence after a slow start, collecting back-to-back boundaries off fast bowler Jerome Taylor, pulling to midwicket and driving through the covers.
Williamson then hit Benn for two fours, pulling through mid-wicket and then on-driving.
Unbeaten on 40 at lunch, the right-hander added just three on resumption before perishing in the third over after lunch, caught by Gayle at slip stabbing at one from Benn that bounced. He faced 68 balls, batting 109 minutes and hit seven fours.
Twenty balls later, Benn pulled off a stunning catch at gully, diving to his right to come up with a one-handed catch to send back Taylor for 45, as the batsmen fended a short one from Roach.
Taylor faced 71 balls in just over 1-1/2 hours at the crease and struck seven fours.
With New Zealand tottering on 114 for four, Neesham and captain McCullum came together in a fifth wicket stand of 54, which arrested the slide.
Neesham, always looking for scoring opportunities, faced 91 balls in 137 minutes at the crease and struck ten fours and a six - a clean blow over wide long-on off Taylor.
He was off the mark with a streaky boundary past slip off Benn, but blossomed to play confidently especially against the seamers.
He flicked Roach off his pads for four before steering Taylor to the third man boundary and driving the same bowler crisply through mid-off for another four in the same over. He reached his half-century in the fourth over before tea, with three to third man off Benn.
McCullum, meanwhile, scored at better than a run-a-ball and struck five fours - three of which came off a poor over from Roach. The first two boundaries were hustled through point while the third was clipped through mid-wicket.
However, McCullum became Benn's third scalp, attempting to turn one that bounced into the leg-side and top-edging a catch to wicketkeeper Denesh Ramdin.
New batsman Bradley-John Watling made just one before falling in Benn's next over, caught at slip by Gayle as he pushed forward defensively, and when Tim Southee was bowled by Benn for six, New Zealand were languishing at 194 for seven.
Neesham and Craig, however, defied the Windies in a 64-run, eighth wicket stand, taking their side to tea at 217 for seven, and then continuing to frustrate the hosts on resumption.
Craig faced 93 balls in nearly three hours at the crease, striking four fours and a six - a fierce hit over long-on off off-spinner Shane Shillingford.
West Indies seemed clueless about breaking the stand when Neesham gifted his wicket, hesitating over a sharp single to cover and then failing to beat the throw at the non-striker's end.
Neil Wagner held out for half-hour before glancing a leg-side catch to Ramdin at 277 for nine to fall for two and Trent Boult smashed a four and a six in 12 before missing a lunge at Benn, and being stumped.