Police said in a statement that a hoax device was found strapped to the body of the man killed by armed officers.
One man was being treated for life-threatening injuries after being stabbed, a woman suffered less serious wounds and another woman was hurt by glass after an officer fired his weapon, the statement from London’s Metropolitan Police said, Reuters reported.
Some witnesses said the man had been armed with a machete and one said he was wearing silver canisters on his chest.
Footage filmed on the high street in the Streatham district of south London showed two men in plain clothes - one in a hoodie and one with a police baseball cap - pointing their guns at a body on the pavement nearby.
“The incident was quickly declared as a terrorist incident and we believe it to be Islamist-related,” said Lucy D’Orsi, deputy assistance commissioner in the Metropolitan Police.
Police said the situation had been contained.
Sky News said it understood that the man shot dead was “under active police surveillance” and had been released from prison in January.
“An investigation is taking place at pace to establish the full facts of what happened,” Prime Minister Boris Johnson said in a statement. He said his government would announce plans on Monday for making changes to the system for handling people convicted of terrorism offences.
The last such incident in London was in November, when police shot dead a man wearing a fake suicide vest who stabbed two people to death and wounded three more before being wrestled to the ground by bystanders.
That attack, at London Bridge, was carried out by a man with extremist militant sympathies. He had been jailed for terrorism and released early.
London’s mayor Sadiq Khan said in a statement after Sunday’s incident: “Terrorists seek to divide us and to destroy our way of life - here in London we will never let them succeed.”