Police were called at 22:33 BST on Wednesday to reports of a man injuring people with a knife.
One of the victims, a woman in her 60s, was treated at the scene but was pronounced dead a short time later.
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 the events.
Scotland Yard’s assistant commissioner for specialist operations, Mark Rowley, said one woman and four men had suffered various injuries in the attack.
The Met’s homicide command was leading the investigation into the attack, with support from the counter-terrorism unit, he said.
He went on: “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.”
A police forensics tent has been erected on Southampton Row, outside the park in the centre of the square, which is close to the British Museum.
There is a heavy police presence in the area and the initial cordon around the crime scene has been extended, according to BBC reporter Andy Moore at the scene.
The condition of those injured and the extent of their injuries is not known at present.