Anyone flying into the country from overseas will have to show proof of a negative Covid test before setting off.
It comes as a ban on travellers from South America and Portugal came into force on Friday over concerns about a new variant identified in Brazil.
Boris Johnson said the new rules would be in place until at least 15 February.
A further 1,280 people with coronavirus have died in the UK within 28 days of a positive test, taking the total to 87,291.
The latest government figures also showed another 55,761 new cases had been reported - up from 48,682 the previous day.
Meanwhile, more than two million people around the world have now died with the virus since the pandemic began, according to figures from Johns Hopkins University.
Speaking at a Downing Street press conference, the prime minister said it was "vital" to take extra measures now "when day by day we are making such strides in protecting the population".
"It's precisely because we have the hope of that vaccine and the risk of new strains coming from overseas that we must take additional steps now to stop those strains from entering the country."
All travel corridors will close from 04:00 GMT on Monday. After that, arrivals to the UK will need to quarantine for up to 10 days, unless they test negative after five days.
Boris Johnson, who said the rules would apply across the UK after talks with the devolved administrations, added that the government would be stepping up enforcement at the border and in the country.