While the seamers bowled well, inviting edges from the batsmen, it was Santner who made scoring difficult. His patience paid off, as Sibley chased a wide delivery, which spun viciously and took the batsman's outside edge through to the keeper.
Burns, who earned a reprieve on 31 after he was dropped by Tim Southee at deep mid-wicket, failed to make the most of it, top-edging a sweep to give Santner his second wicket of the day, an ICC report said.
England slipped further as the left-arm spinner dismissed Jack Leach in the final over of the day for a 12-ball duck. Leach, who was caught by Latham at short leg, decided against reviewing his wicket, but replays confirmed that there was no bat was involved. His wicket helped extend New Zealand's dominance over the game as they ended the day with a lead of 207 runs.
Earlier, England bowlers were brilliant in the first hour of the day, giving little room for the hosts to prosper. New Zealand could manage only 19 runs in the first 16 overs as Stuart Broad and Jofra Archer bowled with discipline. However, once New Zealand negotiated the first hour cautiously, runs started coming freely.
Santner, in particular, was impressive, upping the ante with some incredible shots over the fence. He was especially ruthless against the spinners, taking on Leach and Joe Root. He brought up his half-century just before lunch. At the other end, Watling assumed the role of an anchor, batting through tough periods with immense skill and confidence.
If the first session was slow-moving, the second proved to be a complete opposite, as the seventh-wicket pair unleashed a flurry of shots to boost the scoring rate. The hosts crossed the 500-run mark in no time, with Santner closing in on a maiden Test hundred. He reached the landmark with a pull to fine leg off Ben Stokes in the 187th over, inviting applause from his teammates and the crowd.
The achievement only egged him further as he drove and pulled to take the lead past 200. He was eventually dismissed on 126 by Sam Curran, who once again came to England's rescue, but not before Santner had set a record with Watling for the highest seventh-wicket stand in New Zealand's Test history.
Watling, meanwhile, continued to bat responsibly and showed his class by becoming the first New Zealand wicket-keeper to score a double ton in Tests, nudging a ball from Root behind square for a quick single. He finally fell for 205, after a gritty knock that put the hosts on top, as they with a massive lead of 262.
New Zealand - 615/9
England - 353 & 55/3 (UNI)