New Zealand Prime Minister John Key said it had been "a privilege" to work alongside Mr Cameron.
"He has been a strong leader for Britain, and I'm proud to call him a friend.
"I look forward to working with him during his final months in office."
On the decision to leave the EU, Mr Key said: "This was always a decision for voters in the UK and we respect the decision they have made.
"We will continue to have a strong relationship with both the EU and the UK, and to further develop our ties with both. In this respect nothing has changed.
"The UK remains a member of the EU for the moment and it will take some time to work through the implications of their decision to leave," he said.
"In terms of our existing trade arrangements, the immediate effects of the leave vote on New Zealand are likely to be limited and we expect that trade and other business activities will continue smoothly in the interim.
"We remain committed to the launch of formal negotiations on an EU FTA, and will be working with the UK as they go through the process of leaving the EU to put in place new trading arrangements," Mr Key said.
Labour leader Andrew Little says the New Zealand government should move quickly to deal with Britain on issues such as free trade and immigration in the wake of the Brexit vote.
Mr Little said the Brexit vote in the referendum in the UK would lead to a period of uncertainty but all was not lost.
"For New Zealand it's about how quickly we can get talks underway for direct trade relations and no doubt other matters as well, including movement of people.
That should be priority No 1 for the Government."
He said it was also important to ensure that free trade negotiations with the rest of the EU stayed on track. New Zealand was on the verge of starting those negotiations and the referendum result means it will lose one of its most important supporters in Britain. The free trade talks will inevitably be delayed as Britain goes through the process of extricating itself - which is expected to take at least two years.
NZ First leader Winston Peters - a Brexit supporter - said New Zealand had nothing to fear from the result. "I think the scaremongering will be exposed for what it is: pure scaremongering."
He said it meant the United Kingdom was now free to deal with New Zealand in areas such as trade and the result would strengthen the relationship immensely.
Mr Peters compared the British voters who voted to leave the EU as showing "the same character they showed when they confronted Hitler".
A Brexit supporter, he said the victory for the Leave campaign in the referendum in Britain was a "massive wake up call" for the British establishment and democracies everywhere.
"I joined the Leave campaign at the Leave campaign's request and I am delighted the British people exhibited the same character they showed when they confronted Hitler."
He said the Remain campaign had more money and the advantage of political incumbency, but British voters had stood up to it.
"The people who pay the taxes and face some of the greatest competition for jobs, working conditions and a decent economic and social future simply are not going to go on being ignored or treated as invisible."
Mr Little had also said he would have preferred Britain to stay in the EU, but it was a democratic decision which could not be ignored. He said there was an element of "anti-elite" thinking about it and voter rebellion at being told what to do.
"But I'd say it's more about a widespread concern about the fact that after so many years of being part of a free trade area that many people feel as if they're missing out. Whether it's an immigration thing or a share in economic growth thing generally I think the reality is a lot of people feel it's not working for them."
He said the result did not bode well for British PM David Cameron who had called the referendum and campaigned to remain with the EU.
"Obviously the majority of his fellow citizens didn't feel the same way so his leadership and authority is heavily undermined by the result."
Almost two thirds of New Zealand's trade was with the United Kingdom in the 1950s but that declined after Britain joined the EU in 1973. Since then, New Zealanders' rights to live and work in the United Kingdom have gradually eroded to compensate for increased migration from Europe.
Sir Don McKinnon, former Commonwealth Secretary General, said British voters who believed the Commonwealth could replace the EU in a post-Brexit world would be sorely disappointed.
"There are certainly a group of people in the UK who wrongly believe that the Commonwealth can become the alternative. The Commonwealth can never be a multi-trading group of countries because they have such vast differences in their standards of living, costs of living, and everything else. That is something that will eventually come to rest and people will realise you can't just replace Europe with the Commonwealth."
Sir Don said while it was true New Zealand could secure a free trade agreement with Britain separately, it would not have the benefits of a wider EU agreement. The result meant New Zealand had lost an important ally in Britain in its free trade talks with the EU.
"We don't have that ally now. An independent trade agreement with Britain might be a good thing, but of course it is just a 60 million person market as opposed to a 500 million person market in Europe."
He said only three percent of New Zealand exports currently went to the UK and a further three per cent to the EU.
He said membership of the EU had brought a lot of benefits British voters appeared to have forgotten about, such as freedom of movement around the EU. "This is going to take a long time to sort out."
He said the result was a bit of a surprise to him, although the polls had been close. "Uncertainty is a big thing. We just don't know how this is going to play out as Britain untangles all those linkages with Europe."