After securing an 84-run first innings lead, the Black Caps overcame a difficult start with the bat to extend it to 204 at the close on 3-120.
Rachin Ravindra made a valuable 60 not out, sharing an unbroken fourth-wicket partnership of 69 with Daryl Mitchell to put New Zealand in command on Saturday.
Openers Tom Latham and Devon Conway – centurions in the first innings – both went cheaply second time around.
Jofra Archer did the damage, trapping Latham lbw for four with the final ball of the opening over and then inducing Conway (5) to edge to first slip after unsettling him with a blow to his helmet. His menacing opening five-over burst yielded 2-12.
Gus Atkinson then removed Henry Nicholls for 16 with Harry Brook taking a sharp slip catch.
But Ravindra and Mitchell calmly steadied the ship, leaving England in a tight corner after their first innings had fallen away disappointingly.
Resuming on 2-223, they subsided to 354 all out with Nathan Smith - New Zealand's leading wicket-taker in the series with 14 – taking advantage of a surface offering far more help to the bowlers than on the first two days to take 4-91.
Without Matt Henry and Kyle Jamieson and having lost Blair Tickner concussed by a short ball from Archer, the rest of the New Zealand attack had to step up and they did.
Smith was well supported by Will O'.Rourke who took 3-53 and Tickner's replacement Zak Foulkes, who finished with 3-35 including the key wickets of Ben Stokes and Harry Brook, who made 58 from 80 balls.
It was, emphatically, New Zealand's day as they seek to inflict what would be a seventh defeat in nine tests on England.
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