The blaze took hold from the ground floor to the sixth floor of the flats in De Pass Gardens, Barking, at about 3.30pm (UK time), forcing residents to evacuate.
About 100 firefighters worked for more than two hours to subdue the fire, which is now extinguished. The cause is being investigated.
Two people have been taken to hospital with minor injuries.
Nearby flats have also been evacuated and the Met has put road closures in place.
Difficulties with water supply, resident says