The man has been named locally as Bryce Hodgson, The Mirror reported, and is said to have appeared in court last year on stalking charges.
Metropolitan Police said it was called to reports of a man attempting to force entry to a residential property in Bywater Place, Rotherhithe, around 4.55am on Tuesday.
The man was threatening to hurt the people inside, and was said to be armed with weapons including a crossbow, said police.
Among weapons found at the scene were crossbows, a knife, a sword and a hatchet, the Independent Office for Police Conduct (IOPC) said.
Unarmed officers tried to speak to the man but were threatened, and when police marksmen arrived they entered the building and he was shot.The man was given first aid but died at the scene. His next of kin have been informed.
Met Assistant Commissioner Matt Twist said officers on the scene "had genuine fears for their safety" and had also attempted to get the man to surrender.
"This is an extremely difficult day for everyone involved," he said.
"I would like to pay tribute to the bravery of all our officers who responded and thank all the emergency service staff involved."
The IOPC, which is investigating the shooting, said the dead man was known to at least one of the people at the address.
The police watchdog confirmed two shots were fired from "a police issue firearm" and that the dead man was wearing body armour.Two people in the building suffered minor injuries.
A "detailed search" of the scene continued on Tuesday evening and the IOPC carried out "house-to-house enquiries in the neighbourhood", which are expected to continue in the coming days.
"We attended the police post incident procedures where the officers involved provided their initial accounts of the incident," the IOPC said in a statement.
"HM Coroner has been informed and a post-mortem examination will take place."
More than one crossbow was found at the scene, as well as a gas mask, stab vest, jeans and jacket that were strewn at the scene as well as crossbow bolts, a knife with a holster, a sword, an axe and military style boots.
The IOPC is investigating the incident, in line with standard practice for a fatal police shooting.
Its regional director Mel Palmer said: “Our thoughts are with all of those affected by this tragic incident. The man’s next of kin have been informed and, in due course, we will be contacting them to explain our role. We will keep them regularly updated throughout our investigation.
“Given a man has died after being shot by police, our role is to independently investigate all of the circumstances surrounding this incident including the actions and decisions taken by the police.
“We appreciate that the community will want answers quickly and our investigators are working hard to establish the facts and piece together what occurred.”
Footage shared with the Standard shows police outside the house at the time of the incident. Shouting, believed to be from police officers, tells someone to “get out of the house”. A voice is heard saying “everyone take positions of cover".
Neighbours described the "terrifying" incident. One man who lives just yards from the scene, said: “There was a man shouting, then a woman, there was someone banging.
"Police turned up, they were armed and it was total chaos. We were absolutely terrified.”Another said: “I heard what I thought were shots. It was pitch black then there were blue lights everywhere.”
Another resident Valeria Bocanera said: "There was a man and a woman shouting then it just went quiet. Then the police came. It was horrible.”
One father told how his young daughter had heard a gunshot, and told him "Papa, I heard someone get shot."
“I heard a man slamming on a door - it woke me up," he told the Independent. "It was like he was trying to force his way in. Then the police car arrived with sirens. It was about 5am. I couldn’t hear what he was shouting but he was trying to get in.”
A police cordon and white police tent were in place outside Bywater Place on Tuesday morning. Photos showed forensics specialists and uniformed officers at the scene.
Police are carrying out door to door inquiries on the close knit estate.
Bywater Place is a residential area in the former docklands area of Rotherhithe, around 110 metres (360ft) from the River Thames. It lies around three-quarters of a mile (1.2km) from Canada Water Underground Station.
Adam Hood, who represents Surrey Docks ward on Southwark Council, said he was "deeply concerned" by Tuesday's incident. "Details are still emerging but at this point I'm thankful to the fast response from emergency services," he wrote on X, formerly Twitter.
There were collecting any CCTV and doorbell footage from around the area.
In a statement issued on Tuesday morning, a Met police spokesperson said: "It was reported that the man, believed aged in his 30s, was armed with weapons including a crossbow and was threatening to harm residents inside the address.
"Local officers attended and attempted to speak to the man but, after being threatened, armed officers were called and quickly attended the scene.
"The man had got inside the property, and a police firearm was discharged.
"Officers and paramedics from the London Ambulance Service immediately provided first aid but sadly the man died at the scene.
"Two occupants received minor injuries during the incident.
"The Met’s Directorate of Professional Standards has been informed. A referral has been made to the Independent Office for Police Conduct (IOPC) which has begun an independent investigation."
Detective Chief Superintendent Seb Adjei-Addoh, responsible for policing Southwark, said: “My thoughts are with all those affected by this incident.
"I understand the local community will be concerned at the events that have taken place this morning. We will fully support the IOPC investigation into the full circumstances of what happened."
Valeria Bocanegra, who lives nearby the scene, reported hearing a man and woman "screaming" in the early hours of Tuesday, before later looking out and seeing police at the scene.
"I feel afraid," she told MyLondon. "I thought it was a secure area but not any more, you never know.” She described Bywater Place as "a very nice community" where neighbours know one another.
A man died after being shot by armed officers responding to calls for help from occupants of an address in #Southwark.
We understand the local community will be concerned by this incident and the matter has been referred to the #IOPC
Read more 👇https://t.co/KTAZFxuoJ7
— Southwark Police | Central South BCU (@MPSSouthwark) January 30, 2024
Fatal police shootings are relatively rare in the UK. Just three took place in the year leading to March 31 last year.
Prior to Tuesday's shooting, the most recent Met fatality was the death of Giedrius Vasiljevas, 40, who was shot at his home in Dagenham, east London, in November.
The father-of-four had called 999 to say he had loaded guns and wanted to take his own life.