D'Orsi said police believe the bloodshed was related to Islamic extremism.
Their conditions are now not known.
Up to 180 convicted Islamist terrorists have been released early from jail in the past two decades, analysis shows, and news reports had warned Amman was due for release in the wake of November's London Bridge attack.
The Metropolitan Police said the man died at around 2:00 pm (1400 GMT) on Streatham High Road, a busy thoroughfare lined with shops in a residential neighbourhood.
"We have an issue about if you have about 200 or so inside, the question is how do you detain them?" But now the incident has been fully contained and there are no additional attacks.
Scotland Yard said armed officers were following Amman on foot as part of a "proactive counter-terrorism surveillance operation" in Streatham High Road.
Another woman in her 20s who suffered minor injuries, believed to have been caused by broken glasses triggered from the discharge of the police firearm, continues to receive treatment at hospital, the Met Police said.
Amman reportedly pleaded guilty to six counts of possessing material useful for terrorism and seven counts of disseminating material on Skype between April 19, 2017, and April 2, 2018, on the basis of recklessness.
He said had it not been for the brilliance of the emergency services, or if the wounds being an inch either side of their impact, "we could be having a conversation today about a number of people having lost their lives as a effect of terrorist attack that needn't have occurred". "It could have been much worse than it was", a Whitehall source said. Stop!" Tahir said. "But he didn't stop, and then I saw that they shot him three times.
Police later said that the assailant was killed and there were "two injured victims".
"If someone is clearly driven by an ideology and they believe that slaughtering other people is a sort of God-given objective, then I can see a case for that", said Sir Mark.
Ministers are braced to face immediate demands for all terrorists who have been released early to be recalled, amid growing fears over public safety following two attacks by terrorists on parole within just over two months.
"Terrorists seek to divide us and to destroy our way of life - here in London we will never let them succeed", Khan said in the statement.
"I'm going to level with you", Johnson began, in response to a journalist who asked him if the public could be sure that terror convicts were coming out of prison less risky than when they went in.
BBC journalist Steffan Powell, in Streatham, said armed police were everywhere and had been telling residents to stay inside.