The US will use the UK bases for the “specific and limited defensive purpose” of destroying Iran’s missiles “at source”, Sir Keir said.
He added that the move was in accordance with international law, and the UK government has published what it said was a summary of its legal advice.
It was reported last month that the British government had not given permission for the US to use UK military bases to support any potential US strikes on Iran.
US President Donald Trump said it “may be necessary” for the US to use RAF Fairford and Diego Garcia for operations should Iran “decide not to make a deal”.
Both bases have been used by the US in the past to conduct long-range heavy bombing missions.
The US and Israel began attacking Iran early on Saturday, with Trump encouraging the Iranian people to remove the country’s government.
The Iranian regime has responded with attacks on US assets and countries in the region with a US military presence, including Bahrain, Qatar, the United Arab Emirates, Kuwait and Iraq.
On Saturday, Sir Keir said UK aircraft were “in the sky” in the Middle East as part of a defensive operation to protect its allies and citizens in the region, but said the UK “played no role” in the strikes.
In a video statement posted on social media on Sunday evening, Sir Keir said British aircraft had successfully intercepted Iranian strikes, but added: “Our partners in the Gulf have asked us to do more to defend them and it’s my duty to protect British lives.”
He began the statement by saying it “remains the case” that the UK is “not involved in the strikes on Iran”.
At least 200,000 British citizens are in the region – including residents, those on holidays and passengers in transit, Sir Keir said – and the government “will continue to do all we can to support” them.
British people, including members of the armed forces, as well as allies, were being put at “huge risk” from Iranian strikes, he said, accusing the regime of “becoming even more reckless”.
Iranian strikes have “hit airports and hotels where British citizens are staying”, and on Saturday “hit a military base in Bahrain, narrowly missing British personnel”, Sir Keir said.
He added: “The only way to stop the threat is to destroy the missiles at source, in their storage depots, or the launchers which are used to fire the missiles.
“The United States has requested permission to use British bases for that specific and limited defensive purpose.
“We have taken the decision to accept this request to prevent Iran firing missiles across the region, killing innocent civilians, putting British lives at risk, and hitting countries that have not been involved.”