Health Secretary Matt Hancock said he has "great hopes" for next summer but also hopes that restrictions can be lifted from the spring.
The UK is the first country in the world to start using the Pfizer vaccine after regulators approved its use last week.
Mr Hancock told BBC Radio 4's Today Programme that the vaccination programme will make a huge difference to restrictions.
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He said: "We've said that we think that, from the spring, things can start getting back to normal, and, because we've been able to get this vaccination programme going sooner than anywhere else in the world, we'll be able to bring that date forward a bit.
"I have great hopes for summer 2021 and I hope we can lift the restrictions from the spring."
Mr Hancock said the vaccine rollout is an "incredibly important moment on the march out of this pandemic, but we've still got a long march to go this winter".
He added: "People need to keep respecting the rules and try to live in a way that if you have the virus infects as few people as possible.
"And we are seeing rising numbers of cases in parts of Essex, parts of Kent and parts of London in particular - and we've got to keep this under control."
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The Health Secretary also signalled face coverings would no longer be mandatory in the way they are now in places like public transport, shops and pubs - with £200 fines for those who don’t wear a mask and don’t have an exemption, reports the Mirror.
But he said some people have “got used to” wearing masks and people would have to decide what to do based on “personal responsibility”.
He also said people should carry on wearing masks for now - even if they've received the Covid-19 vaccine.
He told LBC Radio: “You still ought to because we don’t know the impact of the vaccine on you transmitting the virus. You may still have the disease asymptomatically and still transmit it. And that’s one of the things we will be measuring as we roll out the vaccine.”
