On a day that belonged to the Black Caps, with recalled batsman Peter Fulton blasting his second century of the match, England made a disastrous start to the daunting run chase on a turning Eden Park wicket.
By the close of play they were batting for survival with any hope of an unlikely victory long gone, and even a draw looking improbable.
For England to win, they would have to post the most successful fourth-innings chase in Test history. The current record is held by the West Indies, who scored 418 for seven to beat Australia at Antigua in 2003.
But that never looked likely as Tim Southee claimed Nick Compton for two in the second over and Jonathan Trott followed for 37, caught behind off Neil Wagner, to have England at 60 for two.
Captain Alastair Cook, who was dropped on one, set about consolidating the innings with Ian Bell as they put on 30 runs in 25 overs.
But in the dying moments of the day occasional tweaker Williamson claimed two wickets in four balls as Cook went for 43 and nightwatchman Steven Finn failed to score.
Williamson had the remarkable figures of two for five off 6.1 overs.
Bell, intent on preservation, was not out eight after facing 89 balls in nearly two hours -- in sharp contrast to the lusty batting spree earlier by Fulton and Brendon McCullum who mocked the England attack.
They belted 117 runs in a 101-ball partnership as they raced to put New Zealand into an imposing position before declaring at 241 for six midway through the afternoon session.
The 34-year-old Fulton, after posting 136 in the first innings, scored 110 in his second turn at bat to become only the fourth New Zealand batsman, and the first in 22 years, to score back-to-back centuries in a Test.
New Zealand resumed the day at 35 for three and cracked 141 off 26 overs in the morning session.
Dean Brownlie, who had done a sterling job with Fulton to resurrect the innings after New Zealand were eight for three on Sunday, was the only wicket to fall before lunch.
He fell to an exceptional catch by Bell who ran 25 metres from mid-on to clasp the ball in a desperate lurch which left him sprawled on the ground.
McCullum then joined Fulton to belt the England attack around the small ground with Fulton, who was dropped on 31, showing no sign of the nervous 90s as he brought up his century by smashing Stuart Broad over his head for six.
It was the fifth time he had cleared the ropes in an innings that also included 14 fours.
McCullum was 67 not out in an innings that included three sixes and five fours when he declared with the dismissal of BJ Watling for 18.
For England, Monty Panesar took two wickets, although his 9.2 overs conceded 53 runs while Broad took two for 54 off 17 overs.
The first two Tests in the three-match series were rain-affected draws.