Arlene Foster has said her party is not calling for a hard Brexit and is against any physical infrastructure on the border.
Speaking in London yesterday, the DUP leader said she wanted Northern Ireland to have a strong and positive relationship with both the Republic of Ireland and the EU.
She also stressed she had no problem with the Irish language and did not see it as threat to the Union or her unionism.
British values of "diversity and tolerance" meant there was a place for Irish culture in Northern Ireland, she said.
Mrs Foster was addressing what the DUP described as a business event.
She said she wanted Brexit completed "sooner rather than later" and in a common sense way.
"Some have asked if a sensible Brexit means we want a hard Brexit or if we prefer a soft Brexit," Mrs Foster said.
"When we say we want a sensible Brexit, we want to see the referendum result respected.
"We want all of the UK leaving the EU, thus giving us control of our laws, our money and our borders, but doing so in a way that achieves the best possible outcome for us.
"In Northern Ireland, we want to see no physical infrastructure on our border with our Republic of Ireland neighbours.
Mrs Foster said the DUP continued to work with the Government "to make sure the border issue is not used as an excuse to stop or water down" Brexit.
She re-stated her party's opposition to a border in the Irish Sea.
The DUP leader told her audience of her policeman father being shot by "the cowards of the IRA" because "he was proud to put that uniform on and defend democracy against terror".
Bullets and bombs could not "dampen our unionism and our Britishness," she said.
"I am hugely proud to be British, but our Britishness is about much more than the passport we hold. It cannot and should not be reduced down to a name or a badge," Mrs Foster stressed.
"It is about a shared history going back generations. Pride in a UK which ended the slave trade, was the home of the industrial revolution and which founded the welfare state.
"What knits us together isn't a common political creed, one religion or the same skin colour.
"We are bound together by a set of common values, like democracy, freedom, respect for the rule of law and tolerance for others."
Mrs Foster said while she would never permit "a situation where Irish is forced on anyone or becomes equivalent to English", she didn't see the language as "a threat to my unionism or the maintenance of the Union".
She continued: "I would be being utterly selective in my Britishness if I were to argue, as I do, that our diversity and tolerance of it is what makes the UK a success but at the same time say that there is no place for Irish culture."
Referring to the approaching centenary of the creation of the Northern Ireland state, Mrs Foster said it was "an anniversary many on all sides" may have thought would never happen.
"The bulk of our first 100 years has seen us either in actual conflict or arguing over our constitutional status. The next 100 years need not be the same," she added.
While Northern Ireland had benefited financially from the Union, the DUP leader's support for it didn't depend on economic arguments. The relationship of Northern Ireland people to the UK could not be judged just in terms of "pounds and pence", she said. "It is measured in the blood sacrifice at the Somme and Messines and across Flanders fields during the Great War," she said.
"It is measured in the enrichment of our cultural life made by writers like C. S. Lewis and Seamus Heaney."