Mitchell Santner scored his maiden test century and then grabbed three quick wickets to reduce England to 55-3 at the close of play on the fourth day of the first test at Bay Oval on Sunday.
England are still 207 runs behind New Zealand's first innings of 615-9 declared and need to bat the entire final day in Mt. Maunganui on Monday to save the match.
Joe Denly was on seven, while Joe Root will join him on Monday after nightwatchman Jack Leach was caught in close for a duck off Santner with two balls remaining in the day's play.
Left-arm spinner Santner had earlier removed openers Dom Sibley for 12 and Rory Burns for 31. He finished the day with figures of 3-6 off 8.4 overs.
"It's not going to be easy to take seven wickets," Santner said in a pitchside interview of the possibility of completing victory on Monday to take a 1-0 lead before the second match of the two-test series starts in Hamilton on Nov. 29.
"The ball is getting older and getting softer.
"We just have to stay in the areas for longer (to put pressure on England's batsmen) and go from there."
Captain Kane Williamson had closed New Zealand's first innings with a 262-run lead about 15 minutes after the tea break with wicketkeeper BJ Watling having scored his first test double century.
Watling was eventually dismissed for 205 and, having batted for just over 11 hours, he did not take the field for the first hour of England's second innings with Tom Latham taking the gloves temporarily.
The 34-year-old Watling, however, took a sharp catch when he did return to dismiss Sibley off Santner. The catch was a milestone in itself, with the last 101 wickets taken by New Zealand at home all falling to pace bowlers.
Todd Astle was the last spinner to take a wicket at home, against England in March 2018.
Santner struck again with less than three overs remaining when Burns mistimed a sweep and spooned a catch to Colin de Grandhomme at square leg, bringing Leach to the wicket before he fell in the final over.
The 27-year-old Santner had earlier also scored his maiden test century as he combined with Watling for a 261-run partnership, a New Zealand record for the seventh wicket, as the hosts dominated the fourth day's play.
"It was nice to get a hundred (as) it set the team up for a good score," Santner said.
"I think the way me and BJ built the partnership, we knew that if we could bat a long time and get a bit of a lead ... we knew the pitch was going to get harder to bat on.
"It's probably a little bit up and down with those cracks so we just knew that we had to keep building the partnership and keep going."