Led by seamers Matthew Potts and Gus Atkinson, the tourists fought back with the ball after openers Tom Latham (63) and Will Young (42) put on 105 for the opening stand, having been asked to bat.
However, the hard early work was undone by a flurry of loose shots in the afternoon as six wickets fell for 89 runs at Seddon Park.
Late hitting from Mitchell Santner handed some momentum back to the home side, including a straight six off the day's final ball to raise a half-century.
Santner will resume on 50 not out, alongside the scoreless Will O'Rourke.
England's goal of completing a 3-0 series clean sweep was kept alive by some disciplined seam bowling.
Potts (3-75) impressed by removing top-scorer Latham and danger man Kane Williamson (44) to celebrate his recall at the expense of Chris Woakes.
Atkinson (3-55) took his career tally to 51 wickets, second only to Australian seamer Terry Alderman's 54 wickets in 1981 for the most Test scalps in a debut year.
Atkinson was among those to benefit from New Zealand's reckless middle-order batting.
Rachin Ravindra (18), Daryl Mitchell (14), Tom Blundell (21) and Glenn Phillips (5) were all caught by fielders on the off-side when failing to keep the ball down.
It reflected a do-or-die approach, with 228 runs - or 72 percent of their total - coming through boundaries.
Earlier, Latham and Young unearthed some batting resolve largely missing in big losses in Christchurch and Wellington to cede the series at 2-0 down.
Young fell for his lunch score, caught smartly at second slip by Harry Brook off Atkinson after scoring 40 of his 42 runs from boundaries.
It ended the best opening stand of the seam-dominated series from either team, surpassing the previous best of 18.
Latham edged down the leg side off Potts soon afterwards, having earlier survived difficult dropped catches, both by Ben Duckett at third slip when on 12 and 53.
The key wicket of Williamson (44) fell soon after tea, playing on off Potts in agonising fashion, failing to kick the ball clear as it ballooned onto his stumps.
Veteran bowler Tim Southee, playing his 107th and final Test, delighted his home crowd with a whirlwind 23 off 10 balls.
The 36-year-old bowler struck three sixes to take his career tally to 98 and closer to becoming the fourth player to clear the ropes 100 times in Tests.
