Sam Curran (1/26) had Jeet Raval (0) leg-before with his second ball of the innings. Chris Woakes (1/8), coming on somewhat early second change, bowled one that took off awkwardly, taking the edge of Tom Latham (18) en route. New Zealand, 101 behind on first innings, were 34/2 at this stage.
But Kane Williamson (37*) and Ross Taylor (31*) took control, batting till stumps to erase the deficit by all but 5 runs. New Zealand need only a draw to win the series, and with Henry Nicholls, BJ Watling, and Daryl Mitchell to follow, they should be able to achieve that once they see that first hour off.
England started the day on 269/5, 107 behind New Zealand's first-innings total. Rory Burns reached his hundred but fell immediately afterwards, for 101, thanks to an excellent throw by Raval from deep mid-wicket.
Tim Southee (2/90) then had Ben Stokes (26) caught at slip, while Neil Wagner (5/124) got debutant Zak Crawley (1) caught behind. England were 262/5 at this stage.
But Joe Root (226, his career-best) marched on, eventually finding support in Ollie Pope (75). The pair added 196 in just over four hours as the pitch seemed to ease with the day. Once the partnership broke, however, Wagner finished off the tail quickly. England lost their last 5 wickets for 21.
Kane Williamson (New Zealand): Williamson missed the T20Is, and is yet to do play that one defining innings in the Tests. With a Test to be saved and a series to be won, New Zealand will expect their captain to rise to the challenge.
Stuart Broad (England): Broad had hit the right spot consistently in the first innings, and as a result was the most successful of the pack. England's most experienced bowler will have to play a role for them to level the series.
New Zealand 375 (Tom Latham 105, Daryl Mitchell 73; Stuart Broad 4/73, Chris Woakes 3/83) and 96/2 (Kane Williamson 37*; Chris Woakes 1/8) trail England 476 (Joe Root 226, Rory Burns 101; Neil Wagner 5/124) by 5 runs.
New Zealand to bat through the day and clinch the series.