The home side dismissed England for 143 in Hamilton on Sunday and responded to be 3-136 at stumps on day two for an overall lead of 340.
England have already secured series honours after big wins in Christchurch and Wellington but were on the back foot from the very start of a hot day at Seddon Park as they tried to finish off New Zealand's first innings.
Spinner Mitchell Santner (76) and seamer Will O'Rourke resisted for more than an hour to add 32 runs to the overnight tally for the last wicket, taking New Zealand to a healthy 347 all out.
O'Rourke led the way with a brilliant spell as the pair then teamed up with Matt Henry to rip through England's batting order in a session and a half before tea to give the hosts a first-innings lead of 204.
New Zealand declined to force the follow-on and the batters took over for the final session with skipper Tom Latham chopping on after making 19 and Will Young scoring 60 from 85 balls before pulling a Ben Stokes delivery to midwicket.
O'Rourke returned to the crease as nightwatchman but edged behind off Ben Stokes for a duck, leaving Rachin Ravindra to take New Zealand to stumps on two not out in the company of Williamson, who was unbeaten on 50.
New Zealand bowler Matt Henry ripped through the England innings taking 4-48 in Hamilton. (AP PHOTO)
"I think a little bit in the first two tests, we probably didn't have heaps of the rub of the green," said O'Rourke.
"It was big for us just to be able to bring their bowlers back, and having to bowl big spells, and hopefully we can keep keep bringing them back tomorrow."
New Zealand are desperate for a consolation victory to send Tim Southee out a winner on his home ground in his final Test but it was the younger quicks, who will assume the mantle he has worn for 16 years, that did the early damage on Sunday.
Henry (4-48) removed the England openers in the fifth over of the innings, taking a brilliant low catch to remove Zak Crawley for 21 before trapping Ben Duckett plumb in front for 11 four balls later.
O'Rourke (3-33) stepped up after lunch with two wickets in two balls, having Bethell caught at point for 12 before danger man Harry Brook chopped on for a golden duck.
Pope saw off the hat-trick ball but Joe Root had no such luck when an O'Rourke delivery reared up into his face and the batsman cut it to Young at gully to depart for 32 with England having lost three wickets for five runs.
Two in the over for Matt Henry! First an excellent caught and bowled to dismiss Zak Crawley, followed by Ben Duckett LBW for 11. Follow play LIVE in NZ with TVNZ DUKE, TVNZ+, Sport Nation and The ACC. LIVE scoring | — BLACKCAPS (@BLACKCAPS) https://t.co/gATDuNgLhk #NZvENG #CricketNation pic.twitter.com/D0t9RpKFuGDecember 15, 2024
Pope and Stokes put on 52 for the sixth wicket to rescue England from 5-82 but spinner Santner (3-7) came on for the first time and had the wicketkeeper caught in the slips for 24 with his fifth delivery.
The left-armer also sent back the England captain lbw for 27 in his next over before he and Henry tag-teamed to remove Brydon Carse, Gus Atkinson and Matthew Potts cheaply as the England tail folded.
"It hasn't been our best day, it's as simple as that. We're always pretty honest in that dressing room," said England assistant coach Paul Collingwood.
"One thing I would say is that I think we witnessed a pretty special hostile spell of test bowling (from O'Rourke) today. It was great to watch."
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