England wicketkeeper Matt Prior scored a belligerent century to help his side to 315 for nine, having long given up on chasing the 481 required for victory.
The draw ensured that the three-match series, which was affected by rain, ended locked in a 0-0 stalemate.
"We came here to win the series so we are disappointed we haven't done," Cook told reporters at the end of the match at Eden Park. "We are disappointed that we have not played well enough to win the test series. That is the bottom line.
"We fought hard, especially in Dunedin coming back from behind there. Wellington we played a bit better. Here we haven't played as well as you need to do it in international cricket."
The result at least ensured England retained the number two ranking in the world by the cut off date set by the International Cricket Council of April 1.
Had they lost in Auckland they would have dropped to third, and their prize money would have reduced from $350,000 (£230,500) to $250,000 (£164,700).
"We just kept losing wickets at intervals," Cook added of the way New Zealand had kept up the pressure on the visitors throughout the third Test.
"Two last night... obviously two today either side of lunch and then the two just at the end didn't help the nerves in the dressing room.
"But Matt Prior was just outstanding after that to see us home and working with Broady (Stuart Broad) and Belly (Ian Bell), (who) went out there for six hours.
"It was a good situation with the senior players standing up and delivering."
- Sports & Recreation