Health officials said one of the injured was in critical but stable condition, one was stable and the third had less serious injuries.
The attack raises difficult questions for Britain’s government and security services.
Police said Khan was convicted in 2012 of terrorism offences and released in December 2018 “on license”, which means he had to meet certain conditions or face recall to prison.
Several British media outlets reported that he was wearing an electronic ankle bracelet.
Prime Minister Boris Johnson said he had “long argued” that it was a “mistake to allow serious and violent criminals to come out of prison early”.
“It is very important that we get out of that habit and that we enforce the appropriate sentences for dangerous criminals, especially for terrorists, that I think the public will want to see,” he said.
Johnson, who chaired a meeting of the government’s COBRA emergency committee late Friday, said more police would be patrolling the streets in the coming days “for reassurance purposes”.
Police said they were treating the stabbings as a terrorist attack and were not actively looking for any other suspects.
The violence erupted less than two weeks before Britain holds a national election December 12. The main political parties temporarily suspended campaigning in London as a mark of respect.
Metropolitan Police Chief Cressida Dick said officers were called just before 2pm local time to Fishmongers’ Hall, a conference venue at the north end of London Bridge.
The pedestrian and vehicle bridge links the city’s business district with the south bank of the River Thames.
Minutes after the stabbings, witnesses saw a man with a knife being wrestled to the ground by members of the public on the bridge before officers shot him dead.
One video posted on social media showed two men struggling on the bridge before police pulled a man in civilian clothes off a black-clad man on the ground. Gunshots followed.
Another depicted a man in suit and overcoat holding a long knife that apparently had been taken from the attacker.
Karen Bosch, who was on a bus crossing the bridge, said she saw police “wrestling with one tall, bearded man” and then heard “gunshots, two loud pops”.
Another bus passenger, Amanda Hunter, told the BBC that the vehicle “all of a sudden stopped and there was commotion and I looked out the window and I just saw these three police officers going over to a man”.
“It seemed like there was something in his hand, I’m not 100 per cent sure, but then one of the police officers shot him.”
Mayor Sadiq Khan praised the “breathtaking heroism of members of the public who literally ran towards danger not knowing what confronted him”.
“They are the best of us,” Khan said.
London Bridge station, one of the city’s busiest rail hubs, was closed for several hours after the attack.
Scores of police, some armed with submachine guns, ushered office workers and tourists out of the area packed with office buildings, banks, restaurants and bars. Staff in nearby office blocks were told to stay inside.
As police cleared the streets, staff in shops and restaurants ushered customers into storerooms and basements.
Some had been through similar traumatic events in June 2017, when eight people died in the van and knife attack launched by three people inspired by the Islamic State group.