The home side opted not to enforce the follow-on despite having the option after bowling the Black Caps out for 174 and reached 116-1 at stumps as they look to set an unreachable target.
Cook was fluent in reaching an unbeaten 88 off 132 balls but under-pressure Nick Compton was dismissed for seven while Jonathan Trott laboured to 11 not out.
Having been bowled out for 354 early in the day, England initially allowed New Zealand to reach 55 without loss. But Steven Finn put his hand up by taking the first three wickets and Graeme Swann ripped out the middle order with four of his own, including three in the space of seven balls.
New Zealand's embarrassment was partially averted by an aggressive last-wicket stand worth 52.
Conversely, the first 25 minutes of the day unfolded as if scripted by New Zealand. Danger man Matt Prior added just one to his overnight score of 38 before nicking Tim Southee to slip, while tail-enders Finn and James Anderson fell in quick succession to complete Trent Boult's five-wicket haul.
All in all, England managed just 17 runs for their last three wickets. If that gave England's seamers cause for optimism, the opening exchanges successfully dampened it. Where Southee and Boult found consistent swing, the new ball simply refused to move for Anderson and Broad. After 10 overs Peter Fulton and Hamish Rutherford reached 34 for nought, sharing six boundaries.
The New Zealand 50 came up in just 74 balls, a sign of how innocuous England's seamers had been in the period. When they did break through it was a surprise, Finn finding just enough bounce to take Fulton's leading edge. The ball looped in the air and provided a simple return catch.
England showed good control after tea, Stuart Broad providing the control and Finn hunting more wickets. It took him less than two overs to succeed, bringing a ball back into Ross Taylor and bowling him off the inside edge.
It seemed that New Zealand's last hope expired when McCullum was ninth out for 20, waving the bat at a Broad delivery that demanded more respect. But Boult and Neil Wagner put on a dashing half-century for the last wicket.
Boult (24 not out) stunned England by launching Swann for three sixes down the ground, while Wagner took 18 from a single Broad over before Anderson bowled him for 27.
Cook opted not to send New Zealand in again, frustrating some fans who sensed blood, and he batted superbly in hitting 13 boundaries as he closed in on a 25th Test century. But Compton struggled for 45 balls before a bat-pad off Williamson saw him on his way.
Trott's scoring rate was also 15 and frustrated onlookers and although England are firmly in command, a negative approach combined with the prospect of rain on Tuesday could deny them a 2-0 series victory.
- Sports & Recreation
- New Zealand
- Nick Compton
- Graeme Swann
- Jonathan Trott
- Steven Finn