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.
New Zealand had been strongly placed at 97-2 in the 21st over, chasing England's 234 all out in 50 overs which was grafted on a slow pitch offering extreme turn and frustrating big hitters. Ali, who narrowly missed a hat trick, and Rashid shared five wickets to set in motion a collapse which saw New Zealand slump to 103-6 before it ended 230-8.
Williamson hit a ball from Chris Woakes straight back down the pitch and Woakes, on his follow through, deflected the ball onto the stumps after Santner had left his ground. New Zealand was then 199-7 and its hopes of winning appeared over.
But Williamson kept the hosts in the game as he reached his century from 133 balls. New Zealand came to the 49th over, needing 23 runs from the last 12 balls with two wickets standing.
He was able to take only eight runs from the 49th over, bowled by Tom Curran, and New Zealand needed 15 runs to win for the last six balls bowled by Woakes. After a dot ball and a two, Williamson hit the third ball of the over for six to leave New Zealand seven runs from victory with three balls remaining.
Woakes bowled superbly, allowing a two from the 4th ball, then denying Williamson a run from the last two balls to give England a narrow win and a 2-1 lead in the five-match series.
The pitch at the Wellington Regional Stadium, a problem in recent limited overs internationals because of its indifferent quality, was especially poor on Saturday. A dirty, mottled brown, it had been treated with manure in an effort to promote grass growth.
Jason Roy struggled to 15 from 26 balls, Jonny Bairstow hit 19 from 39, and Joe Root a much faster 20 from 22 balls.
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.
But he also found scoring difficult as Williamson, back after missing a match with a hamstring strain, used his slower bowlers well through the middle of the innings. Colin de Grandhomme, bowling his middling medium pace, took 1-24 from 10 consecutive overs. Leg-spinner Ish Sodhi took 3-53, turning the ball prodigiously at times.
"It was a brilliant game, very tight in the end, but it was a brilliant last few overs by us," Ali said.
"It was a great wicket for spinners to bowl on...and the wickets just came really. They weren't great balls or anything but the balls in between were building the pressure.
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