The tourists were on 5-168 at the close on Saturday, leading England by 137 when rain brought a premature halt to the day's play.
New Zealand got off to a promising start in their second innings thanks to a fluent half-century from Tom Latham (76) but lost their way after the opener was dismissed by debutant Jamie Overton (1-34) off the first ball after tea.
Devon Conway (11) was soon to follow, succumbing to Joe Root's part-time off-spin before England landed the big scalp of opposition skipper Kane Williamson, who perished for 48 after edging behind to Matthew Potts (2-20).
Jack Leach (1-26) also got in on the act after Henry Nicholls (7) chipped one straight back to him as New Zealand collapsed from 1-125 to 5-161.
Earlier, Jonny Bairstow smashed a magnificent 162 while Overton fell three runs short of a century as England were dismissed for 360 in their first innings for a 31-run lead.
Resuming on their overnight total of 6-264, Bairstow and Overton continued to play positively early in the morning session before the debutant edged paceman Trent Boult to Daryl Mitchell at first slip to miss out on his landmark.
Bairstow and Overton came together with England at 6-55 and their brilliant rescue act resulted in a 241-run partnership for the seventh wicket to keep the hosts afloat.
Overton struck 13 fours and two sixes in his 136-ball knock.
His dismissal brought Stuart Broad to the wicket and he struck a 36-ball 42, targeting fellow pace bowler Boult in the 60th over by smashing him for two fours and a massive six and punishing Neil Wagner in the next over.
Broad was then bowled by Tim Southee before Bairstow, who hit 136 in 92 balls at Trent Bridge in the second Test to win the match, fell looking to raise the scoring rate as Michael Bracewell had him caught at long-off by Boult.
Bairstow later took the gloves after regular wicketkeeper Ben Foakes was unable to take the field due to stiffness in his back. England said Foakes will be assessed overnight.
England have a 2-0 lead and are aiming to win every match of a home series with at least three Tests for the first time since their 4-0 rout of India in 2011.
