One victim was hospitalized with life-threatening wounds, while the others suffered less-serious injuries in the attack, which took place about 2 p.m. in the Streatham section, a combined residential and commercial neighborhoood about 5 miles (8 kilometers) south of central London, police said.
Karker Tahir said he was at work when he saw police chasing a man down Streatham High Road, the area's main shopping district.
"They kept telling him, 'Stop! Stop!'' Tahir said. "But he didn't stop, and then I saw that they shot him three times. It was horrible seeing it. The man was on the floor and it looked like he had something, which police said may be a device. Police came to us and said, 'You have to leave the shop because he has a bomb in his bag.'"
Images shared on social media showed a man's body on the sidewalk outside a pharmacy. Emergency medical vehicles quickly filled the street and helicopters hovered overhead.
"The circumstances are being assessed,'' the Metropolitan Police tweeted. "The incident has been declared as terrorist-related."
The drama far from central London marked a departure from recent terror attacks in the British capital that took place near high-profile landmarks such as London Bridge and Houses of Parliament.
Investigators gave no immediate details on the attacker. But police declared the violence terrorism-related more quickly than they have in similar cases in the past, suggesting they may have had information about him.
Bell Reberio-Addy, a member of Parliament who represents Streatham, said police assured her it was an "isolated incident."
Stephen Roberts, a former deputy commissioner for the Metropolitan Police, said authorities would quickly try to search the suspect's home and seize any computer devices to scour his internet records. Investigators will be researching the man's background to learn more about the attack and any possible conspirators, he said.
Roberts said if the facts show the stabbings were carried out by a "self-starter" — someone acting alone — it means that any city in the country is vulnerable to a similar sort of low-tech attack.
David Lowe, a former counterterrorism police officer, said reports suggesting the attacker may have been wearing a fake suicide vest indicate he may have wanted to die at the hands of police during the assault.
The bloodshed took place a little over two months after two people were stabbed to death near London Bridge by a man who had recently been released from prison, where he was serving a 16-year sentence for plotting a terror attack.
In November, British authorities lowered the national terror threat level to "substantial,'' meaning an attack is considered likely. That is the third-highest level in a five-step system used by British authorities and marked the first time since August 2014 that the threat level had been so low.
Prime Minister Boris Johnson praised police and emergency workers. London Mayor Sadiq Khan urged community resolve in the face of another attack.
"Terrorists seek to divide us and to destroy our way of life,'' he said. "Here in London we will never let them succeed."