After leaving Number 10 on Tuesday, Ms Foster said the prime minister is set on seeking a deal which "works for the entire United Kingdom" as well as the Republic of Ireland.
In a statement, she said: "We want to see the referendum result implemented. Those blocking Brexit are causing uncertainty but more worrying they are damaging democracy by ignoring the United Kingdom's decision.
"A sensible deal, between the United Kingdom and European Union which respects the economic and constitutional integrity of the United Kingdom, is the best way forward for everyone. We were encouraged by the tone and language in Dublin on Monday.
"History teaches us that any deal relating to Northern Ireland which cannot command cross-community support is doomed to failure.
"That is why the Northern Ireland backstop is flawed. Not one single unionist MLA in the Northern Ireland Assembly supports it.
Mrs Foster went on to say that the PM had rejected a Northern Ireland only backstop in a letter to Donald Tusk on August 19.
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"It is undemocratic and unconstitutional and would place a tariff border between Northern Ireland and the rest of the United Kingdom. That would be unacceptable," she said.
"During today's meeting, the Prime Minister confirmed his rejection of the Northern Ireland only backstop and his commitment to securing a deal which works for the entire United Kingdom as well as our neighbours in the Republic of Ireland."
Mrs Foster emerged from Number 10 with the party's Westminster leader Nigel Dodds after more than an hour.
Mr Dodd told reporters: "It was a very good meeting as always."
"A very good meeting, a very good meeting," he added.