Fifteen fire engines and around 100 firefighters were called to a flat fire on Grantham Road in Manor Park.
Two flats on the 13th and 14th floors were destroyed by the blaze and part of the roof of the 15-storey building was also damaged. Around 60 people left the building before the Brigade arrived. One man was assessed on the scene by London Ambulance Service crews.
The Brigade was called at 1447 and the fire was under control by 1646. Fire crews from Ilford, Stratford, Leytonstone, Plaistow, Dagenham and surrounding fire stations were at the scene.
The Brigade's 999 Control Officers took 53 calls to the blaze.
The fire is believed to have been caused by the accidental ignition of textiles.