"We will collaborate on cyber-related research and development activities, both within government and with the private sector in both countries," the statement says.
Mr McCully also announced a cost-cutting measure whereby New Zealand would offer "hosting arrangements'' to the UK's High Commission in the Solomon Islands.
This was to "return the favour'' for the UK's Kabul embassy allowing New Zealand officials to use its compound.
Mr Hague said this was not about merging diplomatic services.
"We are independent countries, but [it is] about being efficient, about how we work together.''
During their meeting, the pair had also discussed Iran's nuclear programme and agreed the diplomatic process was not over, Mr Hague said.
"I expressed my hope that Iran would come to the next round of talks ready to negotiate seriously. In the meantime, strong and effective sanctions on Iran are very important.''
They also discussed Afghanistan, and Mr Hague expressed the UK's "deep appreciation'' for New Zealand's commitment to the international forces there, and his condolences for those who lost their lives.
Mr Hague would not definitively say whether the UK would support New Zealand's bid to get a seat on the UN Security Council, but gave a strong hint as to his country's position
"As a permanent member of the UN Security Council we're in the position of never declaring our vote in the election so I'm not going to change that position. But, of course, in casting our vote we have always had a strong sense of who our friends are.''
Mr Hague said he was "absolutely delighted'' to be back in New Zealand.
"I attach enormous importance to this relationship. New Zealand and the UK have long and close and excellent relations but we can never afford to be complacent about such a relationship and we have to be clear why it matters; we have to ensure that it adapts effectively to the modern world, and that is the work that we're engaged on together.''
Labour leader David Shearer said his meeting with Mr Hague was "very positive'' and they discussed issues including the situation in the Middle East, and recently imposed restrictions on skilled worker visas for New Zealanders travelling to the UK.
"We used to have 1500-odd people getting skilled visas and going to work in the UK and it's dropped now to about 700-750, whereas about 6000-6500 come from the UK to New Zealand,'' Mr Shearer said.
"It would be good if it could be opened up a bit more. He did say there were other avenues for New Zealanders going and living and working in the UK as well, which he would like us to explore further.''
Mr Shearer said the move to open up New Zealand's Embassy in the Solomon Islands to UK diplomats was positive but further such action should be looked at on a case-by-case basis.
"The UK basing itself in the same premises as the New Zealand Embassy in the Solomon Islands seems to make some sense, in other countries I can see that it might compromise our particular stance on foreign policy.''
Mr Shearer said New Zealand had become more independent in its foreign policy and viewed the world differently to the UK in some instances.
"We mentioned, for example, the issue of Palestinian recognition at the UN. We voted for it and the UK abstained. It's an important distinction between the two of us.
"This [meeting] gives us an opportunity to understand where each other's coming from, from a diplomatic point of view as opposed to a general population point of view.''
While in Auckland, Mr Hague is also scheduled to meet New Zealand business leaders and entrepreneurs, young Pacific leaders, academics, and non-governmental organisations.
Tomorrow he flies to Christchurch to see the quake-damaged city's Red Zone.
It is Mr Hague's second visit to New Zealand in as many years.
