The pair grasped hands but did not smile before sitting down to begin their talks, which took place during the meeting of world leaders in Osaka.
May used the face-to-face session to tell Putin “there cannot be a normalisation of our bilateral relationship until Russia stops the irresponsible and destabilising activity,” Downing Street said.
She said: “the use of a deadly nerve agent on the streets of Salisbury formed part of a wider pattern of unacceptable behaviour and was a truly despicable act that led to the death of a British citizen, Dawn Sturgess.”
The prime minister was also clear with Putin that that the UK has irrefutable evidence that Russia was behind the attack.
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“She said that this behaviour could never be repeated and that the UK wants to see the two individuals responsible brought to justice,” Downing St said.
“The prime minister said the UK would continue to unequivocally defend liberal democracy and protect the human rights and equality of all groups, including LGBT people.”
Her comments came after Putin told the Financial Times that the Salisbury nerve agent attack on Sergei Skripal, a former Russian double agent, last year “wasn’t worth five pounds”.
Ahead of the meeting, the PM said her decision to hold talks with Mr Putin did not mean a return to “business as usual” with Russia.
The Russian president dismissed the Salisbury incident as “fuss about spies and counter-spies” that was “not worth serious interstate relations” and said “traitors must be punished”.
The UK believes Moscow’s GRU military intelligence agency was behind the Salisbury attack on former double agent Sergei Skripal and his daughter Yulia.
Both survived the poison in Salisbury but in July 2018 Dawn Sturgess died after coming into contact with Novichok which is believed to have been in a perfume bottle.
Scotland Yard and the Crown Prosecution Service believe there is sufficient evidence to charge two Russians - known by the aliases Alexander Petrov and Ruslan Boshirov - with offences including conspiracy to murder over the attack on the Skripals.
Online investigation group Belingcat said Boshirov is actually the highly decorated Colonel Anatoliy Chepiga, and Petrov is a military doctor called Alexander Mishkin.
The issue is looming large over the talks between the prime minister and Putin, despite the Russian leader’s attempt to dismiss the incident.
It is the first formal bilateral meeting since the Salisbury attack put the UK-Russia relationship into the diplomatic deep freeze, although the two leaders did speak briefly at last year’s G20 in Argentina.
Earlier May told reporters that the meeting gave her the chance to deliver a “very clear message, leader to leader”.
The UK expelled 23 Russian diplomats May claimed were undeclared intelligence officers following the Salisbury attack and international allies including the US followed suit.
Putin said in an interview with the Financial Times: “Listen, all this fuss about spies and counterspies, it is not worth serious interstate relations. This spy story, as we say, it is not worth five kopecks. Or even five pounds, for that matter.
“And the issues concerning interstate relations, they are measured in billions and the fate of millions of people. How can we compare one with the other?”
He said the security agencies should deal with it, that UK businesses wanted to continue working with Russia, and that he believed the UK and his country were interested in fully restoring relations.
“At least I hope that a few preliminary steps will be made. I think it would be easier for Mrs May, maybe, because she is leaving and is free to do what she thinks is right, important and necessary and not to bother about some domestic political consequences,” he said.
Putin added: “Treason is the gravest crime possible and traitors must be punished. I am not saying that the Salisbury incident is the way to do it. Not at all. But traitors must be punished.”
Ahead of her showdown with Putin, the Prime Minister met Japanese counterpart Shinzo Abe, the summit’s host.
May said: “The UK and Japan are natural partners and we will continue to work together to uphold the global rules by which we seek to ensure a peaceful and prosperous world.”