Theresa May said she would tell her fellow EU leaders that “the Russia threat” did not respect borders, as she arrived for a summit in Brussels.
The UK PM said the nerve agent attack in Salisbury was “part of a pattern of Russian aggression against Europe”.
She will stress the UK’s commitment to European security, after it leaves the bloc in March 2019.
European leaders are also due to decide whether or not to agree the terms for a 21-month Brexit transition period.
Mrs May, who will not be present when the other EU leaders discuss Brexit on Friday, will brief her counterparts on Thursday on the poisoning of former Russian spy Sergei Skripal and his daughter in Salisbury.
The European Council is expected to adopt conclusions condemning the attack, which the UK government has said the Russian state was culpable for – but which Russia denies.
A judge at the Court of Protection in London – Mr Justice Williams – has given doctors permission to take blood samples from the Skripals for testing by the Organisation for the Prohibition of Chemical Weapons.
The pair remain in a critical but stable condition in hospital after they were found unconscious on a bench on 4 March.
Detective Sergeant Nick Bailey, who became ill after responding to the incident, is no longer in a serious condition.
A fourth patient is now being treated as an outpatient – although police will not confirm whether or not they are an officer.
The Russian Ambassador to the UK, Alexander Yakovenko, said Britain had blamed Russia for the poisoning “without any evidence” and his country “cannot accept that”.
He also described as “totally irresponsible” the comments by Foreign Secretary Boris Johnson on Wednesday that likened the public relations gains to President Putin from this summer’s Football World Cup in Russia to Adolf Hitler’s use of the 1936 Berlin Olympics.
“Nobody has the right to insult the Russian people, who defeated Nazism and lost more than 25 million people, by comparing our country to Nazi Germany,” said Mr Yakovenko.
“It goes beyond common sense and we do not think British war veterans, including those of the Arctic convoys, would share this opinion.”
Speaking after the press conference, Mr Johnson said he did “not want to escalate” the situation any further, but that he stuck by his comments “completely”.
Arriving in Brussels on Thursday, Mrs May said: “Russia staged a brazen and reckless attack against the United Kingdom.
“It’s clear that the Russian threat does not respect borders and indeed the incident in (the town of) Salisbury was a pattern of Russian aggression against Europe and its near neighbours from the Western Balkans to the Middle East.”
She added that, as the leaders remembered the victims of terrorist attacks in Westminster and Brussels, she would be “stressing the UK’s unconditional commitment to the future security of Europe”.
The BBC’s Europe editor Katya Adler said the EU summit was the “first test of European support after 12 months of often bad tempered Brexit negotiations”.
She said that the “very strong support being voiced for the UK” will be reflected at the meeting, but EU countries were not united in “being willing to point their finger at Russia” or taking further steps, such as expelling diplomats.
Mrs May will tell EU heads of state that although the attempted assassination of the Skripals took place on UK soil, the Russian threat does not respect borders and it places all European nations at risk.
Russia’s flagrant breach of international law represents a threat to the basis of democracy across Europe, she will add.
“As a European democracy, the UK will stand shoulder to shoulder with the EU and with Nato to face these threats together. United, we will succeed.”
Meanwhile, a senior Whitehall official said Russia had “shown itself to be a strategic enemy not a strategic partner”.
The official stressed that the UK’s response to the Salisbury attack had been carefully calibrated to remain within the law, and that the UK was “not looking for some big confrontation with Russia or regime change”.
Speaking to BBC Radio 4’s Today programme, the UK’s Culture Secretary Matt Hancock said it was “very important to stand up” to Russia over the attack, but the UK must “continue to engage” with the country.
He added: “We continue to engage with Russia because we seek a position where Russia does abide by internal rules and norms [and] where Russia is a grown-up player on the world stage abiding by the international approaches, for instance not using chemical weapons.”
May’s Russia mission faces Brexit test
By Laura Kuenssberg, BBC political editor
At home, Theresa May has taken a firm rhetorical tone, in response to the nerve agent attack in Salisbury.
What the government hopes for in the next 24 hours though is that the other EU nations are willing to come alongside in the same way.
The US, UK, France and Germany signed up to a very strong and uncompromising statement last week.
But the prime minister will use blunt language on Thursday night to try to create a similarly strong and united front among the other 27 EU member states.
Earlier, Mr Hancock told ITV’s Good Morning Britain that the “best response” to the Salisbury attack “would be for England to go to the World Cup, in Russia, and win it”.
The comments came as Russia’s Consul General in Scotland, Andrey Pritsepov, called for Mr Johnson to go, telling the BBC’s Good Morning Scotland programme that the comments “crossed all imaginary and unimaginary lines” of diplomacy.
Labour leader Jeremy Corbyn repeated his calls for a “robust and serious dialogue” with Russia, and hoped for a statement of support for the UK from the EU after the summit.
“We live in one continent ” he said. “We have to have a process where difficulties can be dealt with.”
But Mr Corbyn questioned statements by some members of government, adding: “I’m not sure the language used by some of our ministers is particularly helpful or sensible.”