Ben Stokes sent the Black Caps back to bat after Stuart Broad dismissed them for 209 in the morning session, 226 behind.
But the trail cooled when New Zealand openers Tom Latham and Devon Conway shared an unbroken standing of 128 from 49 overs at Basin Reserve.
It was the home team’s best position in the series and the lack of concrete chances suggested England would have to work hard to turn their lead into victory.
There were a handful of edges that died in front of the sliding cordon and a couple of moves that briefly threatened to put near receivers into play, but otherwise it was a serene existence for the top-tier pairing.
Stokes, who would probably never turn down the chance to put the opposition back given his preference for aggressive tactics, did not bowl before tea.
Broad had put up previous figures of four for 61 after bowling the last three New Zealand batsmen.
Home captain Tim Southee had started the day by hitting hard and often at England’s attack, making his second best Test score of 73. Fifty of those runs came in the morning game, with just 31 balls while cutting three sixes in an over off. Jack Leach and another of an Ollie Robinson bumper.
Their fun was brought to an end by the introduction of Broad, a skier to midwicket immediately making amends for Leach’s fall on the thin leg a ball earlier.
With the door ajar, Broad jumped at his chance to rack up some neat figures, with Tom Blundell nailing one straight to the middle and Matt Henry waving a neck-high delivery. With the innings taking up a modest 53.2 overs, Stokes was keen to turn on the follow-on, England’s first since August 2020.
James Anderson had remained in the reserve, leading the way with the new ball, but both Latham and Conway looked confident as they reached 40 without losing at lunch. There was nothing more worrying than a couple of shoves from Conway, one that briefly interested Ollie Pope in Leg Gully and another from Leach who flew within a yard of Stokes in the Leg Slip.
The afternoon game was even tougher, with 88 added in 30 chanceless overs. Anderson produced the best spell, working hard on defenses but only managing to take a couple of shots that hit the turf frustratingly close to waiting receivers.
Root’s occasional spin was called upon to supplement the attack, with Stokes conspicuous by his absence, and the team went through before both batsmen brought out their half-centuries, Latham on 124 balls and Conway a quicker three.
Robinson looked tired as he stopped a short ball at Latham, who had previously become the seventh Kiwi to 5,000 Test runs, and saw him crash into the ropes.
Despite leading by 98 at half-time, England had questions to answer as they departed.