The man, believed to be in his 20s, was pronounced dead at the scene at Albany Parade, Brentford at about 20:00 GMT on Friday.
An 82-year-old woman was taken to hospital where she remains in a critical condition.
Police have arrested a man on suspicion of murder and attempted murder.
The man, aged in his 30s, has been taken into custody at a London police station. Police are not seeking any other suspect for the stabbings.
A spokesman for the Met Police said inquiries were ongoing to establish the identity of the dead man.