The ban which, also covers the Central American state of Panama and Portugal - due to its strong travel links with Brazil - and the former Portuguese colony of Cape Verde, came into force at 4am on Friday.
Scientists analysing the Brazilian variant believe the mutations it shares with the new South African strain seem to be associated with a rapid increase in cases in locations where there have already been large outbreaks of the disease.
Transport Secretary Grant Shapps described the ban as a "precautionary" move to ensure the vaccination programme rolling out across the UK was not disrupted by new variants of the virus.
"We don't want to trip up at this late stage. We don't have cases at the moment but this is a precautionary approach," he told BBC News.
"We want to make sure that we do everything possible so that vaccine rollout can continue and make sure that it is not disturbed by other variants of this virus."
British and Irish nationals and others with residence rights are exempted from the measure, which was backed by the Scottish Government, though they must self-isolate for 10 days along with their households on their return.
There is an exemption also for hauliers travelling from Portugal to allow the transport of essential goods.
Shadow home secretary Nick Thomas-Symonds said the ban was a "necessary step" but accused ministers of incompetence and "lurching from one crisis and rushed announcement to another".