Three people were taken to hospital following the blaze at the tower block in Charlotte Despard Avenue.
Three more people were treated at the scene, and around 50 residents evacuated before firefighters arrived.
A four roomed flat on the seventh floor of the ten storey building was destroyed by the fire.
London Fire Brigade says it was met with a well-developed fire in a high-rise building when firefighters arrived.
Station Commander Andy Lewington, who was at the scene, said: “Firefighters were met with a well-developed fire in a high-rise building. Crews worked professionally and safely to bring the fire under control as swiftly as possible, preventing the fire from spreading and causing damage to other flats in the building.
“Throughout the incident, our firefighters were supported by their colleagues in the Brigade’s Control Room. Control officers carried out a number of Fire Survival Guidance calls with residents in the building, as crews responded outside.
“One of the Brigade’s 32-metre turntable ladders was used at the scene as a water tower to help fight the fire from above.”
“I’d like to thank our firefighters and staff for their handwork and determination in responding to this incident. I’d also like to thank our colleagues from the London Ambulance Service and Metropolitan Police Service for their support at the scene.
“The cause of this fire is now under investigation by the Brigade’s specialist fire investigators and the Metropolitan Police Service.”
