Police were called on Wednesday to reports of a man injuring people, including a woman in her 60s who died.
A 19-year-old man was arrested at 22:39 after a Taser was discharged by an officer. The Met said mental health was a “significant factor” in events.
London Mayor Sadiq Khan called for the public to remain “calm and vigilant”.
Two other women and three men received various injuries in the attack, Scotland Yard said.
Two from the group remain in hospital while the other three people have since been discharged, the force added.
After receiving treatment in hospital the arrested man is now in police custody.
The Met’s assistant commissioner for specialist operations, Mark Rowley, said the force’s homicide command was leading the investigation into the attack, with support from the counter-terrorism unit.Paul Ó Geibheannaigh, who lives near to the square, said he had seen the body of the woman “on the pavement” and the area was surrounded by a “heavy armed police presence”.
“I haven’t seen a police presence like that in this area since 7/7,” he said, referring to the terror attacks on the London transport system in July 2005.
A witness, who wished to remain anonymous, said he saw the suspect being chased and Tasered.
He said the “guy kept screaming his guts out” and “police were screaming ‘stop, stop, stop'”.
After the man was Tasered, he said, the man was on the ground “for at least 30/40 minutes” while police waited for an ambulance.
Zuhair Awartani was trying to return to a hotel where he was staying when he saw a “dark-skinned man” in his “late 20s” being arrested.
He added: “We heard that there’s been a stabbing going on. The streets were blocked and no-one was allowed through.”Speaking about a motive for the attack, Mr Rowley said: “Of course at this stage we should keep an open mind regarding motive and consequently terrorism as a motivation remains… one line of inquiry for us to explore.”
Mr Rowley added that Londoners would see an increased police presence on the streets, including armed officers, to “provide reassurance and safety”.
BBC News home affairs correspondent June Kelly said police would be looking at the man’s background and his associates and whether he was known to them and health professionals.
Mr Khan has cut short his holiday and will hold a briefing at City Hall at 15:30 BST.
The mayor said his “heart goes out to the victims of the incident in Russell Square and their loved ones”.
He said police were doing an “incredibly difficult job” and the safety of Londoners was his “number one priority”.
The mayor added that police were “working extremely hard to investigate exactly what has happened and to keep the rest of us safe”.
The Met said the attack was not linked with a separate stabbing in Wandsworth, south-west London, on Wednesday, in which one man was killed and two others injured.
The Canadian High Commission said it was investigating reports a Canadian citizen had been injured in the attack.