A first group of embassy employees was already at Tehran airport about to be flown to Dubai, one European diplomat said.
The British diplomats had spent the night in the security of various EU embassies, notably the French mission, the diplomat said.
The evacuation was decided after Iranian protesters, some chanting "Death to Britain", overran Britain's two diplomatic compounds in Tehran for several hours Tuesday, tearing down the British flag and trashing embassy offices.
The protesters were reflecting official anger at Britain's decision last week to cut all relations with Iran's financial sector as part of a raft of new sanctions unveiled in coordination with the United States and Canada.
The storming of Britain's embassy sparked international condemnation, including a strongly worded statement from the UN Security Council.
UN chief Ban Ki-moon issued a statement Wednesday during a trip to South Korea saying he was "shocked and outraged to hear of the incident in Tehran in which demonstrators entered the British embassy, briefly abducted embassy staff and damaged property."
Even Russia -- Iran's closest major ally -- condemned the incursions as "unacceptable"
It took diplomatic police several hours to free six diplomats sequestered by hundreds of protesters inside a building in Britain's diplomatic compound in the north of the capital, the Fars news agency reported.
Inside the embassy in the city centre, several protesters scattered documents and set them alight, witnesses told AFP. One protester was seen looting a portrait of Queen Elizabeth II.
Iran's foreign ministry expressed "regret" over the incident, but some Iranian officials were defiant, blaming the dramatic scenes on Britain's stance towards their country.
Parliament speaker Ali Larijani said the UN Security Council's condemnation was "hasty," state television reported.
He said "a number of students angered by the British government's behaviour" had carried out their actions because of "decades of domineering moves by the British in Iran."
The head of the parliament's security and foreign policy committee, Alaeddin Boroujerdi, told state news agency IRNA: "Iran respects all international laws and the Vienna Convention (on the protection of embassies) and this issue must in no way cause concern for other diplomats and embassies."
He, too, downplayed the storming of the embassy as "a manifestation of the students' high emotions."
Britain, though, called the acts "a very serious failure by the Iranian government." Britons were warned against non-essential travel to Iran and the few in the country were advised to stay indoors.
The United States -- which cut off diplomatic ties to Iran after students stormed its own embassy in 1979, taking 52 Americans hostage for 444 days -- also expressed alarm.
President Barack Obama said the storming of the embassy was "not acceptable" and that "all of us are deeply disturbed."
Secretary of State Hillary Clinton called it "an affront not only to the British people but also to the international community."
Tuesday's demonstration had been organised days earlier, when the Iranian parliament passed a law to expel Britain's ambassador in retaliation for London's new sanctions.
Britain and the United States have been leading Western moves to step up pressure on Iran over its nuclear programme, which they fear is being used as cover for the development of a weapons capability.
The looting of the British embassy and compound came ahead of an EU foreign ministers' meeting on Thursday that is expected to unveil new sanctions against Tehran over its nuclear programme.
Iran has repeatedly denied its nuclear programme has a military component and has warned it will respond to any military attack by raining missiles on Israel and Turkey.
