First Minister Mark Drakeford has said Welsh Labour's position in the European elections was "too complicated".
Labour was third behind the Brexit Party and Plaid Cymru, and criticised for not taking a clear position in favour of a new referendum on Brexit.
Mr Drakeford now says he backs a new poll and remaining in the EU.
Pro-Brexit Tory AM Andrew RT Davies said a new referendum was not the answer to election results that were difficult for the two main parties.
Meanwhile, former Welsh local government minister Alun Davies accused Mr Drakeford of shadowing UK leader Jeremy Corbyn on "absolutely everything" - demanding the Welsh Labour leader take responsibility for Labour's poor election showing.
In his first interview since polling day, the first minister told Claire Summers on BBC Radio Wales he had spoken to Mr Corbyn and explained Welsh Labour's new position.
"I have now concluded that the only way we can try to guarantee a future for Wales that would not be a catastrophe is to put this decision back to the people in a referendum," he said.
"We were doing our best to respect the result of the original referendum.
"People in Wales narrowly voted to leave the EU and ever since what we were trying to do was to find a way of respecting that vote by crafting a form of Brexit that would have gone on supporting the Welsh economy, guaranteeing prosperity here in Wales."
He added things had "changed fundamentally" with the Conservative leadership race "a contest between harder and harder forms of Brexit".
"That's very different from where we were only a week or ten days ago," he said.
"Recognising that fact means we have to re-think how we would approach that in Wales.
"The chance of doing a deal of the sort that we have always advocated seems to me now to be at an end.
"There's no prospect of that happening sadly, and in those circumstances our view is that going back to the people, in what has been a very deeply divided society, asking people for their verdict again is the best way forward."
Mr Drakeford said the party's position in the European elections was "clear but complex".
"Complex, because we were trying to demonstrate that we took the referendum result seriously, while also preparing for what would need to happen if we weren't able to strike the deal of the sort that we have always advocated," he said.
"I understand that that was complex and for lots of people that was a difficult position to grasp."
Mr Davies, South Wales Central Conservative AM, said: "Regrettably, the first minister has buckled at the first sign of discontent from his Labour colleagues who have been in denial ever since the people of Wales voted to leave the European Union back in 2016. That's not leadership."
Brexit Party assembly group leader Mark Reckless said: "This tone deaf response will appear misjudged to voters, many of whom elected Labour members, but last week voted for Brexit again."
However, Vaughan Gething, Mr Drakeford's health minister who has campaigned for a new vote, said Mr Drakeford's interview was a "welcome recognition of the truth that had been staring us in the face".
"We now have Welsh Labour in tune with the vast majority of our members and doing what is right," he added.
Lynne Neagle, Labour AM for Torfaen and a People's Vote campaigner, said she was "really pleased" with Mr Drakeford's support for a new vote, but said it was "a pity that it took so long".
"Our message wasn't just too complicated, it was wrong," she said. "Wrong for jobs, for our children, our public services and wrong for Wales.
"I look forward to the first minister enthusiastically campaigning for our communities to have a final say on Brexit in the weeks and months ahead."
Pontypridd AM and Corbyn supporter Mick Antoniw, a member of the Labour National Executive Committee, said Labour "must call for a new referendum as the only way of resolving the Brexit crisis and avoiding no deal", with Labour campaigning to remain but with a second question on the ballot.
He said a new Tory PM would likely opt for a snap election. "If the referendum is lost and there is a majority vote is to leave, there must be a choice on the ballot paper of 'no deal' or 'customs union' options," he added.
Plaid Cymru's Delyth Jewell accused Mr Drakeford of "a cynical reaction aimed at cauterising the haemorrhaging of support from their party".
In an interview with Newyddion 9's Aled ap Dafydd, former Welsh local government minister Alun Davies said "it would have helped" if Mr Drakeford had committed to a second referendum "before the votes had been cast rather than after they had been counted."
In comments recorded before the Radio Wales interview, the Labour AM for Blaenau Gwent said Labour needed to acknowledge it had gotten Brexit "badly wrong".
"We also need to recognise this is a failure of leadership as well," he said. "We haven't had the party leading our supporters, our members, our candidates."
He said the failure rested in both Cardiff and London. "Mark Drakeford has taken decisions to shadow Jeremy Corbyn's position on absolutely everything," he said.
"So he can't then walk away when the going gets tough. He has to accept responsibility for that and I hope he does accept responsibility for that."
Calling for "clear red water" to reappear between Cardiff and London wings of Labour, he said: "We've been following a line from the leader's office in London which hasn't listened to either to members of the Labour party or what Labour voters are telling us."
Mr Davies, who was sacked by the new leader became first minister last year, accused Mr Drakeford of failing to reach out to those who did not support him in the 2018 leadership campaign.
"I hoped that Mark would reach out after his election to bring people together," he said. "For whatever reason, Mark has chosen not to do that."