The Black Caps extended their overnight 338 for six to 454 for eight declared, scoring 116 runs in just 16 overs on Monday before captain Brendon McCullum called a halt.
That left England needing 455 to win.
The most any side have made in the fourth innings to win a Test is the West Indies' 418 for seven against Australia at St John's, Antigua, in 2002/03.
England's corresponding record is 332 for seven against Australia at Melbourne back in 1928/29.
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Adam Lyth, who scored a maiden Test hundred in the first innings, was 24 not out on his Yorkshire home ground.
England captain Alastair Cook, who in the first innings became England's highest run-scorer in Tests, was 18 not out.
New Zealand were unlucky to be 1-0 behind in this two-match series after scoring more than 700 runs and taking 20 wickets in the first Test at Lord's yet still going down to a 124-run defeat.
Mark Craig, 15 not out overnight, was dropped on 23 when a diving Stuart Broad at mid-off failed to hold a tough chance off James Anderson.