The tourists went into the match requiring a win to keep the series alive following their defeat by three wickets on Sunday.
As it was, the likes of James Anderson and Steven Finn restricted the home side to 269 runs - a total that England made light work of in reply. After the match, Cook admitted that a lot of the praise should go to the bowling unit.
"I thought the way we bowled up front was outstanding," he told Sky Sports News. "Steven Finn and James Anderson really bowled well. We gave them nothing to get going, took some wickets, and put them under pressure. It was only a fantastic effort from Brendon McCullum that got them up to a decent score.
"We knew 270 was probably a little bit below par on that wicket. We knew if we kept our heads batting and kept wickets in hand, as it proved it was quite an easy chase."
The final match of the series will be played on Saturday.