Twelve fire engines and around 80 firefighters were called to a fire on Heathstan Road in Shepherd's Bush.
Balconies on the second and third floor were damaged by the fire as well as a small part of the connecting flats. Part of the flat roof was also damaged. Around 30 people left the building before the Brigade arrived. A baby with smoke inhalation was taken to hospital as a precaution by London Ambulance Service crews.
Station Commander Matt Brown, who was at the scene, said: "Firefighters worked hard in hot conditions to tackle this blaze, which started on a second floor balcony.
"It also affected a third floor balcony and the third floor flat roof.
"Firefighters carried out a systematic search of the building and one of our 32m ladders was at the scene to assist firefighting operations.
"Residents in the area were advised to close their doors and windows due to the smoke.
"While there were no road closures, we asked motorists to please avoid the area where possible as there were a large amount of vehicles at the scene."
The Brigade was called at 1445 and the fire was under control by 1826. Fire crews from Park Royal, Chiswick, Fulham, Willesden, Richmond, Hammersmith and surrounding fire stations were at the scene.
The cause of the fire is under investigation.