The visitors are 303 for two at stumps, 86 runs behind the hosts.
A 148-run stand between openers Tom Latham (59) and Martin Guptill (70) was halted by their dismissals within the space of three balls but Kane Williamson (92) and Ross Taylor (47) sustained the impetus by making 155 in an unbeaten third wicket partnership.
Eight more runs would see Williamson become the 18th New Zealander on the Lord's honours board, and a 10th century before the age of 25 means he would join the group of Don Bradman, Sachin Tendulkar, Alastair Cook, Garry Sobers, Neil Harvey, Graeme Smith and Javed Miandad who have done likewise or better.
Each of New Zealand's top four showed the benefits of getting into line behind deliveries with limited lateral movement off the wicket.
Latham and Guptill's stand was the second highest by a New Zealand opening pair at Lord's, falling short of the 185 set by John Wright and Trevor Franklin in 1990.
Williamson and Taylor's stand was the third highest for the third wicket at the ground, behind Martin Crowe and Bruce Edgar's 210 in 1986 and Bevan Congdon and Brian Hastings' 190 in 1973.
The latter pair eased into their work in favourable batting conditions.
A customary murmur of appreciation came as Williamson delivered consummate on-drives, back foot cover drives and pulls while Taylor played the innings of a professional, working myriad singles off his legs and behind point. His moment of most vulnerability came on 20 surviving an lbw review off Anderson.
The DRS ruled the ball was hitting leg stump, but with more than half missing. Umpire Ravi was vindicated.
The day was not without further drama. In the space of five balls the match spun into a state of flux during the middle session.
Moeen Ali provided the catalyst for change. The off-spinner trapped Latham lbw. Two balls later, in Stuart Broad's over, Guptill blasted a drive to Gary Ballance at cover. Taylor's first ball resulted in a shout for lbw, which Hawkeye suggested was going over, and three subsequent leg byes. Then Williamson dabbed his first ball into the covers for a single which would have seen Taylor run out at the striker's end if Broad had underarmed accurately.
The scenario was a reflection, in miniature, of why this test has been a compelling contest of ebb and flow.
Earlier, New Zealand fought back to dismiss England for 389.
Statistics suggest the visitors would head into rare territory if they won. Across 130 tests and 131 years at the ground, only one team has lost after posting more than 389 on the first innings.
England made 425 in 1930, but succumbed to Australia who replied with 729 for six, including 254 to Don Bradman.
Led by Trent Boult with two wickets, England only added 35 runs on the second morning.
Looking refreshed, Boult attacked the crease to get Ali (58) and Broad (3) caught behind. His efforts continued the momentum gained when he had Jos Buttler adjudged lbw with the last ball of the opening day.
Matt Henry snaffled Jimmy Anderson caught and bowled for 11. That left Boult and Henry one wicket short of a place on the dressing room honours board. Boult finished with four for 79 from 29 overs; Henry took four for 93 from 24.5.
In the pre-match warm-up wicket-keeper B-J Watling failed a fitness test but is expected to bat. His right knee swelled up when a Henry delivery hit on the first day. Latham's makeshift work with the gloves across 74.5 overs proved competent. He conceded 15 byes but took sound catches of Joe Root, Ali and Broad. His batting vigil provided a good advertisement for adjusting to the pace, bounce and slope of the wicket beforehand.
England pace bowler Mark Wood suffered the biggest disappointment when New Zealand were 40 without loss. He lured Guptill into playing at a short length ball which he snicked to Alastair Cook at first slip on 24. The jubilation turned to consternation when a third umpire check revealed a no-ball.
Wood's still searching for his maiden wicket.
Guptill batted with a black armband out of respect to his wife's grandfather who died recently.
