Devon Conway made 200 on his Test debut before England captain Joe Root helped the hosts recover from a top-order collapse on the second day of the series opener at Lord’s on Thursday.
Opening batsman Conway was last man out in a first-innings total of 378.
England, who slumped to 18-2, ended the day on 111-2, a deficit of 267 runs.
Burns was 59 not out and Root unbeaten on 42, with their partnership worth 93 runs.
Earlier, South Africa-born left-hander Conway was in sight of becoming the very first batsman in the 144-year history of Test cricket to carry their bat throughout the entire innings in their first knock in the format when he was run out to end a stay of more than nine-and-a-half hours.
Nevertheless, it needed a review of a close call before it was confirmed Root had taken the bails off in time.
Even so, Conway was only the seventh batsman to make a double hundred on Test debut.
He faced 347 balls, with 22 fours and went to 200 in style when he hooked fast bowler Mark Wood for the only six of his innings.
New Zealand No 11 Neil Wagner, also born in South Africa, made 25* off 21 balls that included a superb straight six off Stuart Broad during an entertaining last-wicket stand of 40.
Sussex paceman Ollie Robinson, like Conway making his Test debut, led England’s attack with 4-75 runs in 28 overs.
But New Zealand, who face India in the inaugural World Test Championship final at Southampton later this month, then struck twice despite being without left-arm spearhead Trent Boult, who had been on family leave following a stint in the Indian Premier League.
Towering paceman Kyle Jamieson had Dom Sibley lbw for a duck with a ball that straightened the decision upheld on ‘umpire’s call’ after a review.
But there was no room for doubt when Zak Crawley was caught behind off Tim Southee for two.
Burns and Root, however, then held firm in an often attritional stand, although left-hander Burns did go to fifty in 90 balls.
New Zealand resumed on 246-3, with Conway 136 not out after Black Caps skipper Kane Williamson had won the toss.
But they lost four wickets for just six runs as they slumped from 288-3 to 294-7, with Mark Wood, the fastest member of England’s pace quartet, instigating a collapse.
Conway shared a fourth-wicket stand of 174 with Henry Nicholls before his fellow left-hander was caught holed out to long-leg off Wood for 61 by England debutant Ollie Robinson.
It was a welcome moment for Robinson, who found himself having to apologise after play on Wednesday when racist and sexist tweets he had first posted as a teenager were published again.
Wood then removed BJ Watling for just one, with the aid of a sharp slip catch by Sibley.
Fellow seamer Robinson, who took two wickets on Wednesday, then had Colin de Grandhomme lbw for nought on review, with Mitchell Santner also out for a duck when he chipped Wood to mid-off.
Robinson, however, was deprived of a place on the Lord’s honours board when Broad dropped Southee at mid-off to deny him a fifth wicket.
© Agence France-Presse