After resuming Saturday's third day on 223-2, England collapsed to 354 all out, to ultimately concede an 84-run first-innings deficit.
Nathan Smith claimed 4-91 and Will O'Rourke took 3-53 as the tourists exploited increasing seam movement and uneven bounce.
Harry Brook top-scored with 58, while Jacob Bethell added 74, but England lost wickets regularly after Joe Root, Bethell and Jamie Smith all fell in the opening six overs.
Zak Foulkes, playing as a concussion substitute, impressed with 3-35, including the key wickets of Ben Stokes and Brook.
England responded strongly with the ball as Jofra Archer produced a fiery spell to remove Tom Latham and Devon Conway, reducing New Zealand to 12-2 and briefly reviving the hosts' hopes.
However, Rachin Ravindra's unbeaten 60 and Daryl Mitchell's 26 not out guided New Zealand to 120-3 at stumps with an unbroken 69-run partnership.
Gus Atkinson claimed the other wicket, dismissing Henry Nicholls, as England were left facing an uphill battle heading into day four.
Pakistan debutant Razaullah, batting at number 9, smashed a record-equalling nine sixes to keep his team in the third Test against England at Edgbaston, with home needing a target of 130 to win.
New Zealand white-ball captain Mitchell Santner has declined a lucrative offer from Australian Big Bash League side Adeliade Strikers to remain in contention for Test selection.
England were on 154-4 at stumps on the first day of the third Test against Pakistan at Edgbaston after the visitors once again folded with bat to be 133 all out.
Kevin Pietersen is set to return to the England setup after being appointed as a specialist mentor to the men's white-ball teams ahead of the 2027 World Cup.
