First Minister Mark Drakeford has accused England's health secretary of making "simply factually untrue" comments about Wales' vaccine rollout.
Matt Hancock said Wales had a faster programme because it could draw on England's supply if needed.
But Mr Drakeford said Wales' success was based on the management of its own stocks, not any extra supply from the UK government.
Wales currently has one of the best vaccination programmes in the world.
People aged 25 to 29 in England can book to have their first dose of Covid vaccine, while in Wales nearly two thirds of 18 to 29-year-olds have had at least one dose of a vaccine.
Latest figures on Friday, showed more than 300,000 people aged under 30 in Wales had been given a first doses, joining similar numbers in their 30s.
Mr Hancock told MPs England had a slower rollout due to its "buffer" of supply.
He said UK ministers needed to make sure there was always enough supply for second doses, while Wales could draw on England's supply if needed
But Mr Drakeford told the BBC's Newsnight: "It's simply factually untrue.
"We've drawn nothing out of an English buffer, we manage our own stocks and the fact that we have some of the best vaccination rates in the world is just because of the way in which the programme in Wales has been organised and delivered."
Mr Drakeford also said he was not prepared to follow England by announcing a date when all freedoms would be restored in Wales.
"I think it's too optimistic to say that vaccination has broken the link between falling ill and hospitalisation with the new Delta variant," he said.
"We really don't know that sufficiently to be confident that the vaccination is going to defend us in exactly that way.
"Wales is very largely free already. A great majority of things that people weren't able to do before, they're able to do already."
Welsh Conservative Senedd leader Andrew RT Davies said Wales was "extremely fortunate to enjoy a world-leading vaccination programme" but said it would not have been possible with a Labour government in Westminster.
"Let's make no bones about it. This first-class programme wouldn't have been possible if Labour had been calling the shots as Wales and the rest of Britain would have been stuck in the EU scheme," he said.