Matt Hancock told BBC One's The Andrew Marr Show the UK has 350 million doses on order.
He said the vaccines would still be offered “according to need” targeting the elderly and people with underlying health issues but added: “Every adult will be offered a vaccine by the autumn”.
The government was “accelerating the rollout” of the vaccine and more people had been vaccinated in the last week than in the whole of December, Mr Hancock said.
One third of people aged over 80 had now received a vaccination, he confirmed.
Mr Hancock has said the Government is on course to reach its target of 13 million people vaccinated by mid-February, with 200,000 people being vaccinated per day.
The opening of mass vaccination centres this week is likely to increase the rate of jabs.
Ministers are hopeful that vaccinating the most vulnerable members of the population will pave the way restrictions to be eased.
But Professor Peter Horby, chairman of the New and Emerging Respiratory Virus Threats Advisory Group (Nervtag), does not believe it will mean a return to business as usual.
He told Andrew Marr: “I don’t think it will return us to normal. There will still be a large number of people being infected, and although the absolute risks of someone under the age of 80 dying or ending up in hospital are low, with a large number of infections that still translates into a lot of people and so we’re going to have to manage the virus, with social distancing measures as well as vaccination for the coming months.”
Asked about the likelihood of social distancing measures being in place next winter, he said: “I think that’s likely. I think it very much depends on how well we can scale up the vaccine programme and how quickly we can get it out to a substantial proportion of the population.”
The UK has recorded more than 80,000 Covid-19 deaths since the start of the pandemic, the highest death toll in Europe.
This week people without coronavirus symptoms and who cannot work from home are to be prioritised for quick turnaround tests made available to every local authority.
The most terrifying words in the English language are: I'm from the government and I'm here to help.