Pursuing a world-record chase of 455 had long since been abandoned as England – resuming on 44 without loss – fell to 62-4 on the final day.
Kane Williamson was the unlikely bowling hero with figures of 3-15 as the Black Caps wrapped up victory an hour after tea to draw the two-Test series??1-1.
Cook, who was England’s man of the series, told Sky Sports: “It was a disappointing day today. After halfway we were right in the game but credit to New Zealand.
“We created these little opportunities, I think they were 20 for two, and then they built a partnership – 140 for four – again there was an opportunity there??but every time we had a chance, New Zealand piled on the pressure with attacking batting and took the game away from us.”
Cook, who surpassed Graham Gooch as England’s all-time record Test run-scorer on Sunday, as always found positives in the performance, praising Adam Lyth, who scored 107 in the first innings, Ben Stokes and Mark Wood for their??contributions.
He said: “For Adam to score his first 100 on his home ground was an amazing achievement and he’s looked a really good batter at the top of the order.
“And Ben has taken a big step forward I think – obviously at Lord’s with the headlines – but in general with the way he’s played, even today I thought he??played really nicely in a tough situation.
“And I think Mark Wood as well, he’s come in and has added a bit of fire to our bowling attack.
“There’s a lot of good things, a lot of things which we will go away and wish we would have done differently in this Test match in particular.
“As always against a good side, if you’re not 100 per cent on your A game you get found out so credit to (New Zealand captain) Brendon (McCullum) and his??team.
“In tough situations where this game was in the balance they took the positives under pressure and they deserved the win.”