Ten fire engines and around 70 firefighters were called to a fire at a block of flats on Westbridge Road in Battersea.
A flat on the 20th floor was destroyed by the fire. Two people left the affected flat before the Brigade arrived. A child was taken to hospital by London Ambulance Service crews. A further 50 people left the building before the Brigade arrived.
Station Commander Pete Johnson, who was at the scene, said: "Crews were faced with a lot of smoke issuing from the top of a block of flats on arrival.
"There was also lots of visible flame which has prompted a high number of calls to our Control Officers.
"Firefighters worked quickly to bring the fire under control."
The Brigade's 999 Control Officers took 18 calls to the blaze.
The Brigade was called at 2002 and the fire was under control by 2053. Fire crews from Battersea, Fulham, Wandsworth, North Kensington and surrounding fire stations were at the scene.
The fire is believed to have been accidental and involved a candle.