Officers were called to Shepperton Road in Islington, north London, shortly after 23:00 BST on 5 September.
The Independent Police Complaints Commission, which is investigating the incident, said two shots were fired.
Both the police officer who pulled the trigger and the woman said the 40-year-old man had a knife.
A post-mortem examination found the man died as a result of shock and haemorrhage caused by a gunshot wound to the the left chest.
He also suffered a bullet wound to his upper left arm.
It is the first fatal shooting by a Metropolitan Police marksman since the death of Mark Duggan in Tottenham, north London, in 2011.
During the incident, a police negotiator and firearms officers were called and shortly before 01:00 BST a number of shots were fired at the man by police.
He was pronounced dead en route to hospital.
The man has not been named as police are awaiting formal identification.
IPCC commissioner Cindy Butts said: "I have today met with members of the bereaved family, who are very distressed, to extend my condolences and to update them on the progress of our investigation.
"I would like to assure all those concerned that the circumstances surrounding this tragedy will be investigated thoroughly."
The IPCC is currently conducting door-to-door enquiries and has requested interviews with the officer who fired the shots and other officers who were present at the scene.
The police watchdog is also analysing CCTV and witness statements provided by the police officers.
A man who was killed by police had pinned his ex-partner against a wall at knifepoint when the shots were fired, witness statements have said.