The tourists saw their second innings get off to a dreadful start, with both openers out for a duck, and they were 12 for three when Ross Taylor was dismissed.
At lunch on the fifth and final day, New Zealand were 21 for three, needing a further 324 runs to reach their victory target of 345.
But, having outplayed England for much of this match, it now looked as if they would do well to escape with a draw in the first of this two-Test series.
The second ball of New Zealand's chase saw Martin Guptill caught in the slips by Gary Ballance off Anderson.
And New Zealand were still on nought when Stuart Broad's first ball of the second over had Tom Latham lbw for a golden duck.
For New Zealand, it was all starting to become horribly reminiscent of their last Test at Lord's, in 2013, when they collapsed to 68 all out chasing 239.
And the Black Caps would have been six for three had not Joe Root, at fourth slip, dropped Taylor off Anderson.
Earlier, England resumed on 429 for six, a lead of 295, with skipper Alastair Cook unbeaten on a commanding 153 -- his second Test hundred in as many matches but first on home soil for two years -- after Ben Stokes had struck the quickest Test century at Lord's in terms of balls faced off just 85 deliveries.
But Cook's marathon innings came to an end when, pushing forward to Trent Boult, he got an inside edge and was caught behind by substitute wicket-keeper Tom Latham, although it needed a review to overturn umpire S Ravi's original decision.
His innings extended the left-handed opener's England record for most Test hundreds to 27. And he will now need just 32 runs in the second Test at Headingley to surpass Essex mentor Graham Gooch's England Test runs record of 8,900.
Cook's exit was the start of a Boult burst that saw the left-arm paceman wrap up the innings with four wickets for nine runs in 17 balls.
That meant Boult finished with fine figures of five for 85 that saw him gain a coveted place on the Lord's honours board for the first time in his career.
But England's new-ball burst meant a draw, not a record-breaking win, was now the priority for the tourists.
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