Updating MPs, the Prime Minister said it was “highly likely” that Russia was responsible for what happened.
But Moscow hit back, claiming Mrs May’s statement was “another political information campaign based on a provocation”.
Russian Foreign Ministry spokeswoman Maria Zakharova said there had been a “circus show in the British parliament”.
And Moscow added, cryptically: “Before making up new fairy tales, let the British disclose how the Litvinenko case ended.”
Russian dissident Alexander Litvinenko was poisoned with radioactive polonium-210 in London in 2006.
His widow, Marina, says Britain appears unable to protect those seeking political asylum in the UK.
Later on Monday, Moscow’s foreign ministry said that any British threats to boycott the World Cup, being held in Russia in the summer, would “fan anti-Russian hysteria, complicate relations between our countries and (be a) blow to world sport”.
The poison used to target former spy Sergei Skripal and his daughter Yulia was from the Russian-made nerve agent group novichok, Mrs May said.
But its use does not necessarily mean Russia was to blame, a former Kremlin adviser told Sky News.
Alexander Nekrasoff said it was “possessed by about 16 countries in their laboratories”.
“Why do I know this?” he added. “Because that’s how the antidote is developed.”
He also claimed that the Porton Down military research facility near Salisbury possessed novichok – again in order to develop an antidote.
Mr Nekrasoff described Mr Skripal as being of “no interest to the Kremlin”, and said people in some of the other countries allegedly in possession of novichok “want to damage Russia”.
Vladimir Dzhabarov, a deputy head of the foreign affairs committee in the upper house of the Russian parliament, said the claim of Russian involvement was “nonsense”.
It would not make sense to attack an ex-spy “whom no one needed”, he added.
Mr Skripal, 66, and Yulia, 33, have been in a critical condition in hospital since being found unconscious in Salisbury on Sunday 4 March.
Mrs May said their poisonings took place “against a backdrop of a well-established pattern of Russian state aggression”.
Britain was ready to take “much more extensive measures” against Russia than in the past, she added.
Russia has been given until the end of Tuesday to respond.