Welsh First Minister Mark Drakeford has called for a delay to Brexit after Theresa May's deal was overwhelmingly defeated in the House of Commons.
MPs voted 202 in favour versus 432 against - with the government losing by a historic majority of 230.
Mr Drakeford said it was time for the UK government to change direction and seek an extension to Article 50.
Welsh Secretary Alun Cairns said MPs should say what they want "because we clearly know what they don't want".
Labour leader Jeremy Corbyn said it was a "catastrophic" defeat, and said he would table a motion of no confidence in the government on Wednesday.
But Mrs May signalled her intention to carry on in a statement immediately after the vote.
"The House has spoken and this government will listen," she told MPs.
The prime minister offered cross-party talks to determine a way forward on Brexit, if she succeeded in winning the confidence vote.
An overwhelming majority of Wales' 40 MPs - 32 to six - opposed the deal.
Mr Cairns was joined by Tory colleagues Stephen Crabb, Chris Davies, David Davies, Glyn Davies and Simon Hart in voting in favour of Mrs May's deal.
Two Welsh Conservatives - David Jones and Guto Bebb - joined 28 Welsh Labour MPs and the four Plaid Cymru MPs in voting against it.
Labour's Nick Smith acted as a teller for the Noes so was not counted in the total, while Paul Flynn - who has not been in the Commons for some time due to illness - did not attend.
Reacting to Mrs May's heavy defeat in the Commons on Tuesday, Mr Drakeford said it was "hardly surprising that we find ourselves in this situation".
"The Prime Minister's deal always fell well short of what is needed to protect our economy and jobs, surrendering political influence without any guarantee that our long-term economic interests will be protected," he said.
"We must now avoid careering headlong into a catastrophic 'no deal'," the first minister added.
"We cannot afford to gamble with the future of this country with self-imposed deadlines."
Mr Cairns rejected the first minister's call to delay Brexit, saying: "That's not the plan.
"We need to leave the European Union in the time and demands that have been set out," he told BBC Wales Today.
"Members of parliament need to reflect the demands that people made, even in Wales," he added.
The Welsh secretary said Mrs May was extending an "olive branch" to senior MPs from all parties to find a way forward.
"Clearly we want a deal because that is the best way to deliver a smooth exit from the European Union," he said.
"But it's up to senior parliamentarians to say what they do want, because we clearly know what they don't want."
Mr Cairns urged them to be responsible, and not make demands "without appreciating the wider context of the negotiations."
Conservative backbencher David Jones said: "I don't think there's any doubt now that the withdrawal agreement is dead."
The ex-Welsh secretary and former Brexit minister said the PM should "consider all the options available".
He said Mrs May should go back to the EU and stress the UK would not accept a withdrawal agreement that "constrains British sovereignty or ties Britain to the customs union".
Otherwise, "the way forward has to be a managed 'no deal'," the Clwyd West MP said, claiming it would save the UK money.
But he said it remained to be seen what could get through the Commons, and urged all parties to consider the way forward without "political point-scoring".
Mr Jones had opposed Mrs May in the no-confidence motion brought by Conservative backbenchers last year, but he confirmed he would support the government in the no-confidence motion called by Labour.
"The fact is that Labour's policy on the withdrawal from the EU is completely incoherent," he said.
"There's no way any other government than the present one should be entrusted with the future of the negotiations."
Plaid Cymru's leader at Westminster, Liz Saville Roberts, said the Commons vote was "a clear and decisive rejection" of Mrs May's Brexit deal.
She added that if Labour's motion of no confidence failed, the party should back another referendum.
"The Brexit pendulum is swinging in our favour and all paths lead towards giving the people a final say," she said.
Islwyn Labour MP Chris Evans said Mrs May's deal was "so bad" it had "united the House, by historic proportions, in condemnation".
"Her inability to negotiate, work across the house and downright stubbornness has made us a laughing stock," he tweeted.
"She has failed time and time again - she must now put our country first and resign."