A woman and her two children were rescued from a fourth-floor flat of the block in Ham Close, Richmond at 9.45pm on Saturday.
Tragically a man on the ground floor flat was pronounced dead at the scene by London Ambulance Service crews.
Firefighters tackled the blaze for nearly two hours, using fire escape hoods to rescue the woman and two children.
Two men and a woman from a third flat were also led to safety by firefighters.
A London Fire Brigade spokesman said: “The Brigade was called at 9.46pm and the fire was under control by 2341. Four fire engines and around 25 firefighters from Kingston, Surbiton and Chiswick fire stations attended the scene.
“The cause of the fire is under investigation.”