Gus Atkinson and Matthew Potts spearheaded England's bowling attack as New Zealand reached stumps on day one on 315-9.
Ben Stokes won the toss and chose to field at Seddon Park in Hamilton on Saturday.
That decision appeared to have backfired when Black Caps openers Tom Latham and Will Young put on 105 between them, before the latter was dismissed on 42 as Atkinson made the breakthrough.
Potts (3-75) subsequently dismissed Latham (63), before Rachin Ravindra (18) was sent packing by Brydon Carse (2-78).
Kane Williamson (44) was Potts' second wicket, and that dismissal at the end of the 59th over sparked a mid-order collapse.
Daryl Mitchell (14), the dangerous Tom Blundell (21) and Glenn Phillips (five) fell to Atkinson, Carse and Potts respectively.
Harry Brook took a spectacular catch in the deep to take out Matt Henry from Stokes' bowling, flicking the ball up while he negotiated the boundary rope before securing it when he was back on the right side of the line, with Tim Southee (23) the final wicket of the day to fall for New Zealand, as Atkinson finished play on 3-55.
Mitchell Santner heads into day two unbeaten on 50, with Will O'Rouke at the other end, as New Zealand will look to push on towards the 350 mark.
Data Debrief: England must buck a trend
New Zealand are undefeated in their last four men's Tests at Seddon Park when they have batted first in the match (W3 D1).
In fact, they have not lost at the venue when batting first since March 2012 against South Africa.
Their innings started well, but England showed resilience to fight back, with Potts - selected ahead of Chris Woakes - one of their standout seamers, albeit the excellent Atkinson, who took a hat-trick in the second Test, closed out day one with the best figures.