Williamson finished 112 not out, rebuilding New Zealand's innings after a middle-order collapse provoked by off-spinner Moeen Ali, who took 3-36 from his 10 overs, and leg-spinner Adil Rashid, who took 2-34.
A brilliant 96-run stand with Mitchell Santner put New Zealand back on track, but the latter was dismissed in unfortunate circumstances when a Williamson strike was deflected onto his wicket by Woakes while he was out of his crease.
Woakes, defending 15 runs in the final over, conceded two twos and a six to Williamson and then fired in two dot balls to prevent a New Zealand victory.
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The tribal party won eight of the nine seats it had contested. "It is a very unexpected loss for the party. We should have given them the confidence that we could take the BJP on", a senior leader from Bengal said.
Speaking to BBC Test Match Special, former England spinner Swann said: 'Mitchell Santner almost won that game for New Zealand - he's got the power, Kane Williamson's more of a nudger and nurdler.
New Zealand were left with 66 runs to get in the final 10 overs and 35 in the last five.
Kane Williamson became the fifth-fastest batsman to reach 5,000 ODI runsIn contrast to England's innings, there seemed to be no evidence of the ball misbehaving when Colin Munro was leading the New Zealand chase. The team England is now travelling in New Zealand and as per the schedule, they are playing an ODI series against the Black Caps.
New Zealand's skipper hit Chris Woakes' third delivery for a maximum but could not repeat the feat from the final delivery as his side slipped to a narrow defeat.
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On a snail-slow wicket at Wellington's Westpac Stadium, the Black Caps' bowling attack - particularly seamers Colin de Grandhomme and Colin Munro and spinner Ish Sodhi - had a field day, flummoxing the English. Skittish at first, the left-hander grew in confidence to punish the inaccuracy of Tom Curran and the fit-again Mark Wood.However, after he was brilliantly caught for 49 at short cover by Ben Stokes off Rashid, the Black Caps came close to a complete implosion.Mark Chapman miscued to point, Tom Latham was lbw on review and Colin de Grandhomme holed out to long-on, all from Moeen's off-spin.
Morgan was the first man to show any mastery of the pitch, reaching his 48 from 71 balls with three fours and a six one of only two sixes in the innings. We stumbled a bit in the middle and it hit us, it meant we had to get deeper in the game. Excluding Sodhi, Boult took two wickets and Southee and Grandhomme took one wicket each. Lower down the order, Buttler, who cracked three fours and a six, Moeen Ali (23) and Chris Woakes (16) made decent contributions and propelled the visitors to a competitive total.
"We were outplayed, especially the way they bowled up top and didn't give us much", said Santner.
'Jason said he had his fingers under it and I believe him but on the TV it's not going to look out, ' said England captain Eoin Morgan.
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Some people think I could do three; some people think I could do five; some people didn't think I could do the bench press. Questions trail UCLA's Josh Rosen even though he looks like a franchise quarterback ready to save the day.