The Prime Minister's plan for an open border between Northern Ireland and the Republic of Ireland has set out a basis for progress, DUP leader Arlene Foster said.
Theresa May said it could be achieved via a commitment that the UK's regulatory standards are at least as high as the EU's.
Mrs Foster said: "The Prime Minister has set forward the basis upon which it would be possible to move forward.
"The issues facing both the United Kingdom and the European Union are of fundamental importance and it is vital that we achieve outcomes that are sustainable for the future."
The Prime Minister acknowledged a responsibility to help find a solution.
"But we can't do it on our own. It is for all of us to work together."
Mrs Foster, whose pro-Brexit MPs support the minority Government at Westminster, welcomed the Prime Minister's "clear commitment" that she will not countenance any new border being created in the Irish Sea between Northern Ireland and the rest of the United Kingdom.
The DUP leader said: "Northern Ireland goods must have unfettered access to trade into Great Britain and the same must apply to Great Britain goods entering Northern Ireland.
"Indeed, it is particularly welcome that one of the 'five tests' is strengthening the Union."
She added: "We want to see an outcome that protects the economic and constitutional integrity of the United Kingdom but one that also enables us to have a good trading relationship with our nearest neighbours."
The DUP's leader at Westminster, Nigel Dodds, said it was a "sensible approach" to how no hard border could work.
Nationalist SDLP leader Colum Eastwood said the Prime Minister's commitment to work for a solution was a belated recognition of reality.
He said: "However, that commitment is severely undermined in the same speech when she advocated the very same customs proposals which were widely and comprehensively dismissed last year."
He said the speech again revealed the "internal chaos" at the heart of the Government.
"The truth is that this remains a negotiation between members of the British Cabinet in London rather than a negotiation with the European Commission in Brussels," he added.
"The British Prime Minister is more concerned with balancing the dogmas of her divided Cabinet rather than balancing the interests of our peoples."