Cook eventually made 43 before driving at part-time off spinner Williamson; Dean Brownlie pouched him at second slip.
Expect New Zealand's progress to contain a few struggles on a good batting wicket. They have been unable to bowl a side out twice in the four tests since victory against Sri Lanka in Colombo in November. That lack of success should change tomorrow.
New Zealand skipper Brendon McCullum stacked the slip cordon for his pace bowlers and crowded the bat for spinner Bruce Martin and later Williamson. He must have revelled in the opportunity to deliver the aggression he has so often sought a licence for.
The last overs included eight of the nine fielders placed around the bat. McCullum has persevered with the same bowlers, even with the return to fitness of Doug Bracewell.
They look like repaying his faith. Curiously it's the first time the same New Zealand XI has played together across an entire three-test series.
McCullum declared the New Zealand second innings at 241 for six, adding to a first-innings lead of 239. McCullum was unbeaten on 67 off 53 balls, while opener Peter Fulton hit 110, his second hundred of the test.
Fulton became just the fourth New Zealander to hit two centuries in the same test after Glenn Turner, Geoff Howarth and Andrew Jones.
Having started the day at 35 for three, leading by 274, New Zealand added a further 141 runs at 5.4 an over in the morning session to be 176 for four at lunch.
An England win would represent the highest successful fourth-innings chase in test history. Only four teams have scored more than 400 to win a test. The current record is 418 for seven by the West Indies against Australia at Antigua in 2003.
England have never chased down more than 332 in the fourth innings of a test (v Australia 1928). They've only successfully chased more than 300 three times. The highest winning fourth innings total at Eden Park is 345 (achieved by the West Indies in 1969).