The second and final Test begins in Christchurch on Friday.
It is only the 10th time New Zealand have beaten England in the 102-Test history between the two countries.
“We'll need a hero,” fast bowler Stuart Broad said at close of play.
New Zealand had declared their first innings on 427 for eight, with Henry Nicholls unbeaten on 145 to show the pitch held no demons.
Henry Nicholls reached his half century to be 52 not out with BJ Watling on 18 and New Zealand were 233 for 4, leading England by 175.
Kane Williamson, who scored his first Test century on debut — 131 against India in 2010 — shared the previous New Zealand record of most hundreds with Ross Taylor and the late Martin Crowe.
One of the best batsmen in the game, Kane Williamson is amongst the finest fielders and keeps testing himself, redefining the incredible acrobatics level.
England were certain that the ball had kissed Chris Woakes' hands and wanted Kane Williamson to accept Woakes' call and walk.
England's innings lasted 20.4 overs as they were bowled out for 58. New Zealand only needed two bowlers, Trent Boult and Tim Southee.
Kane Williamson, Trent Boult put New Zealand in command against England in 1st Test, Day One