James Anderson must wait to become the first Englishman to 400 Test wickets, after taking only the first to move to 399 as the Kiwis were dismissed for 220 and went down by 124 runs.
England will head to Leeds knowing they cannot lose this two-match series, thanks principally to the heroics of captain Alastair Cook (162) and the fastest ever hundred at HQ from Stokes as they combined to overturn a 134-run first-innings deficit.
They set the Kiwis a national record 345-run chase, and Stokes (three for 38) helped to ensure they saw their opponents off in 67.3 of a scheduled maximum 77 overs.
Anderson, who last month overhauled Ian Botham’s long-standing record as his country’s most prolific Test wicket-taker, will join even more elite company the next time he strikes.
He could celebrate in any case after England closed out the win, with a spectacular catch by Moeen Ali at third man as Stuart Broad (three for 50) got last man Trent Boult, despite half-centuries from Corey Anderson (67) and BJ Watling (59).
It was England’s leading wicket-taker who put the tourists in instant trouble when he had Martin Guptill edging an outswinger low to third slip off the second delivery of the innings.
Then Broad struck at the other end with more full-length swing into Tom Latham’s pads to see him off lbw and make the total nought for two.
The Kiwi openers had each made half-centuries in the first innings, and shared a stand of 148. Second time round, they were both gone for nought and lasted three balls between them.
Broad soon doubled up with another lbw, Ross Taylor aiming to leg and missing to make the total 12 for three.
But thanks to stoic pair Kane Williamson and Watling, New Zealand recovered – and then after Stokes had his say with ball as well as bat, big-hitting Anderson launched 10 fours and a six in his 44-ball 50.
Stokes took the wickets of Williamson and Brendon McCullum in successive deliveries.
The admirable Williamson erred for once when he poked a low catch to gully off the back foot, and then McCullum was bowled first ball off thigh pad and bat by a very good one that nipped back up the slope.
Anderson and Watling then put on 107 until the latter gloved a catch behind when Mark Wood found extra bounce.
A ‘golden arm’ bonus from Joe Root, straightening one from round the wicket to have Anderson lbw, nonetheless meant England needed only to see off three tailenders with 23 overs left – and once Stokes yorked Mark Craig, the end was nigh at last.
England were earlier bowled out for 478, having resumed on 429 for six.
Boult finally broke Cook’s resistance, after nine hours at the crease but still 32 runs short of surpassing his mentor Graham Gooch’s all-time England Test highest runs haul.
Boult took all four wickets, at a personal cost of just nine runs – finishing with five for 85 and match figures of nine for 164.
Cook was his first victim, given not out caught-behind on the field but the decision overturned by the third umpire who detected an inside edge pushing forward.
The England captain therefore had to go after facing 345 balls, latterly in a seventh-wicket stand of 66 with Moeen.
But Boult made short work of the remainder of the innings, bowling Broad and Anderson with full-length deliveries and Moeen lbw on the back foot in between.
England’s bowlers instantly appeared on the fast track to a win which 24 hours earlier was a supremely fanciful notion – and although New Zealand refused to go quietly, they could not see out a final day lapped up by 20,000-plus of partisan support.