It is New Zealand's first away win against England in 16 years, second at Headingley, fifth away in 54 attempts and ninth overall in 101 tests between the countries.
The result was a tribute to the cadence at which they played after being sent in. They made 804 runs at 4.93 an over and were denied more than a full day's cricket due to inclement weather. Further forecast rain skirted the venue during the final afternoon.
Extraordinarily, New Zealand's ranking will drop from third to fourth, despite losing 2-0 in England two years ago.
Today's victory was another tribute to Brendon McCullum's leadership. Off the field he insisted they could play the same way they did at Lord's, after seeing the test slip from their grasp on the fourth day.
With the victory he edges ahead of Geoff Howarth as the most successful New Zealand captain by virtue of percentage wins. Howarth had 11 from 30 matches (36.67 per cent); McCullum's has nine from 24 matches (37.50 per cent). Stephen Fleming had 28 from 80 matches (35 per cent).
The hosts were reduced to 102 for five at lunch with rain scheduled to enter down the order in the afternoon. The precipitation proved a flaky teammate, seemingly falling everywhere but the ground. England were left on 206 for eight at tea with 35 overs left.
Kane Williamson was the star of the middle session, moving to figures of three for 15 off seven overs, including the dismissals of Alastair Cook lbw for 56 and Stuart Broad bowled for 23. A Cook review showed the ball to be hitting the top of middle. He had Ben Stokes caught behind before lunch for 29.
Earlier, McCullum channelled chess grandmaster Garry Kasparov, Williamson aped slip aficionado Mark Taylor and Tom Latham mimicked expert short leg David Boon.
McCullum marshalled his fielders around the bat like rooks, knights and bishops, particularly for the off-spin of Craig.
Left-hander Gary Ballance was the best example. Two slips, a leg slip and a short leg assumed their positions as Craig zeroed on the footmarks outside off stump.
McCullum would chat to his bowler, move Henry 5m behind square and make sure the batsman was within earshot.
Checkmate loomed.
Ballance eventually went for six from 26 balls when a Trent Boult delivery clipped his pad and cannoned into off stump. Boult also forced Adam Lyth to defend on the back foot outside off to entice the edge.
With England four down, McCullum inserted himself at silly point when Stokes, who blasted the 85-ball fastest century at Lord's, emerged.
The captain was unflinching even when Craig dropped short and the England all-rounder's backlift took a deep breath, tipping beyond the horizontal in prospect at drilling shots through McCullum's solar plexus.
Craig continued a fine test. He supplemented two for 48 from 26 overs in the first innings with three for 73 from 31.5 overs in the second. Those figures included 15 consecutive overs and seven maidens and two wickets from the Kirkstall Lane end in the opening session, as his responsibility grew.
The territory is nothing new. Craig has bowled New Zealand to victory in three of his previous nine tests; on debut against the West Indies in Kingston, against Sri Lanka in Wellington and against Pakistan in Sharjah.
The decision to open the day with Craig was sound. McCullum was looking for him to hit the footmarks outside off stump against England's seven left-handers, and especially Cook, who appeared to have a GPS locator on his off stump to pace bowling across his last three innings.
The catching of Williamson and Latham provided the other New Zealand highlights.
Williamson showed dexterity taking Ian Bell with two hands, low to his left for one.
Latham's David Boon-esque effort at short leg, to remove local hero Joe Root for a duck second ball, was more unorthodox. Root hit the ball into what proved an adhesive sternum. Latham clasped his hands around the gift. The 1763-strong Headingley crowd was reduced to a murmur.
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