Mrs Foster said there are more things to unite than divide the UK and Ireland in phase two of discussions on the split from Europe.
Speaking in Killarney, in the Irish Republic, she suggested leaders across Ireland should work together.
Mrs Foster said she planned to raise the prospect of enhancing Anglo-Irish relations, under the auspices of the British-Irish Council, when she meets Ireland’s foreign affairs minister Simon Coveney.
“Maintaining Northern Ireland’s economic and political status as an integral part of the United Kingdom is absolutely crucial to me and my party,” she said.
“To think anything else would be as foolish as believing that the Taoiseach (Leo Varadkar) or the Tanaiste (Mr Coveney) desired anything other than Irish unity.
“But while we will always battle for our own national interests, we must also battle for our mutual interests.
“And our mutual interests will not end on the day the UK formally leaves the European Union. The United Kingdom may be leaving the EU but the common interests that we share across the British Isles will remain.”
Mrs Foster was speaking at the Killarney Economic Conference in Co Kerry, discussing Brexit.
Her remarks on forging closer ties and mutual respect are a significant departure from the fallout which marked relations between the DUP and the Irish government towards the end of last year.
That came amid both the fractious finish to phase one of the Brexit negotiations and the Irish border question, and Mr Coveney’s aspirations for a united Ireland.
The DUP leader suggested Anglo-Irish relations could be strengthened through the British-Irish Council, set up as part of the Good Friday Agreement to boost co-operation between the UK and Ireland in areas such as transport, the environment and energy.
“The UK exiting the European Union ought not to become a barrier to continued co-operation on issues of ongoing mutual interest,” she said.
“It especially shouldn’t become a barrier when the infrastructure – in the guise of the British-Irish Council – already exists that can allow us to continue to work together as closely as ever on issues of shared interest.”