UK Foreign Secretary David Miliband has described Russia's actions against the British Council as "reprehensible" and a "stain" on the country's reputation.
He said council staff had been grilled by Russian security services on issues including their family pets' health.
Such actions were "not worthy of a great country", he said, reading out EU and US messages of support for Britain.
The council has suspended work at two Russian offices, saying "intimidation" made it impossible to continue.
Tensions have been growing since Russia refused to allow Britain to extradite a Russian businessman on suspicion of murdering a former KGB agent, Alexander Litvinenko, in London in November 2006.
The British Council found itself at the centre of the row when the Russian government ordered it to close its two offices outside Moscow, accusing it of tax violations.
The Russian allegations and demands had been rejected.
But on Thursday the council's chief executive Martin Davidson blamed "a campaign of intimidation against our staff" for his decision to suspend work at its St Petersburg and Yekaterinburg offices.
He said more than 20 Russian staff had been called to attend interviews with the FSB security service and ten more were visited at their homes late at night by Russian tax police.
"The interviews had little to do with their work and were clearly aimed at exerting undue pressure on innocent individuals," he said.
"Our paramount consideration is the well-being of our staff and I feel we cannot continue our work without significant risk to them."
In a statement to MPs, Mr Miliband expressed "anger and dismay" at Russia's actions.
He said the council's Russian staff had been questioned by security services on everything from the British Council's institutional status to the "health and welfare of family pets".
"We saw similar actions during the Cold War but frankly thought they had been put behind us," he told MPs.
"I think the whole House will agree such actions are reprehensible, not worthy of a great country, and contrary to the letter and spirit of the legal framework under which the British Council operates, notably international law"
He said the Russian foreign minister had made it clear that its "attacks" on the British Council "were linked" to the Litvinenko case.
Measures announced last year - including visa restrictions for Russian officials travelling to the UK - "will continue to be administered rigorously", he said.
Work at two British Council offices has been suspended
But he said cultural activities were "entirely separate" and should not become "a political football".
So he had decided not to take similar actions against Russian activities in the UK - such as masterpieces scheduled for show at the Royal Academy - and said the British Council would continue its work in Moscow.
He added: "Russia's actions against the British Council are a stain on Russian's reputation and standing."
For the Conservatives, shadow foreign secretary William Hague said: "We join you in deploring the Soviet-era tactics employed against legitimate cultural contact, tactics which we trust will be counter-productive against the Russian point of view and are deeply offensive from the British point of view."
Liberal Democrat foreign affairs spokesman Ed Davey added: "The Russian government needs to understand that these bully-boy tactics make them look increasingly ridiculous in the eyes of the international community."
HAVE YOUR SAY
It is unfortunate that the British Council which does much good in Russia has been caught in the political crossfire
John Smith, Barnsley
In addition to the staff interviews, the British Council's St Petersburg director Stephen Kinnock was stopped and questioned by Russian police on Tuesday. He was released an hour later.
His father, British Council chairman Lord Kinnock, said the real motivation for the offensive against the council was "deliberate retaliation" for Britain's efforts to secure justice for a Russian murdered in London.
He told the House of Lords Russian nationals working for the council had endured "systematic bullying" and the view of the international community was that attacks on a widely respected cultural organisation for crude political purposes was "absolutely indefensible".
The tensions between London and Moscow dates back to the death of ex-KGB agent Mr Litvinenko, who was given a fatal dose of radioactive polonium 210 in London, in November 2006.
Russia has refused to hand over businessman and former KGB agent Andrei Lugovoi, whom UK investigators suspect of murdering Mr Litvinenko. Britain expelled four Russian diplomats in response and Moscow followed suit.
Mr Lugovoi, who is now a member of the Russian parliament, says he has been framed and denies any involvement in Mr Litvinenko's murder.
Terry Davis, secretary general of the Council of Europe - of which Russia and the UK are both members - said his organisation could "provide a forum to sort out this issue of extradition".