The last 50-over series defeat of the Kiwis on their territory came in 1991-92 and while this latest victory cannot be compared to the historic first successful Test tour of India for 27 years at the end of 2012, also under Cook's captaincy, it is still another notable achievement.
England did it the hard way too, going 1-0 down in the first match of three in Hamilton but recovering to post comfortable wins in Napier and at Eden Park. Both times, it was the new-ball bowling of Steven Finn and James Anderson that put them in control.
Yesterday, Finn took the honours with figures of three for 27. He and Anderson reduced New Zealand to 11 for three, and despite another fine hand from their captain Brendon McCullum (79), the hosts were unable to recover sufficiently to post a testing target.
"It's been quite a long time since we won a one-day series in New Zealand, so we will enjoy tonight," said Cook, after his team had coasted past 185 all out with five wickets and more than 12 overs to spare. "We've shown a lot of character.
"We were disappointed with ourselves in that first game. Then we had two must-win games and we held our nerve and raised our standard."
Many pundits expected England to add this ODI series to their Twenty20 triumph, but Cook stressed how formidable New Zealand can be in their own country.
"We came into this series and everyone was saying, 'you're expected to win quite easily'. We were saying, 'we'll have none of that because it's 21 years since we won a one-day series here and the Kiwis have just beaten South Africa in one'. We can be mighty proud of the way we've gone about our business this week."
Agreeing that Finn and Anderson were the key, Cook added: "I actually thought they were better than in the last game. There's not actually very much the batsmen can do if they bowl like that. They're taking a big punt to take a risk to try to score some runs."
For England, the performances augur well for future assignments, including next summer's Champions Trophy on their own patch.
"If we can bowl like that we're going to be a very hard side to face in English conditions," Cook went on. "Steven Finn and Jimmy set the tone perfectly, and the other guys backed them up really well."
Finn was bowling off his new shortened run-up for the second time, and Cook said: "I think it has helped him get a little bit more rhythm.He would admit himself that at the start of this tour, and at the start of the India tour, he didn't quite have the rhythm he'd have liked. But something's just clicked this last couple of games, and he's bowled beautifully."
McCullum was unable to keep wicket because of an abdominal strain, but expects to be fit for the start of a three-Test series next month.
Cook, perhaps understandably, fancies England's chances. "Certainly we can take a lot of confidence from the way we bowled at their top order," he said. "We know we can put them under pressure again with the red ball."