About 150 people left the apartment block on Elmira Street before the firefighters arrived, The Telegraph reported.
According to the London Fire Brigade, the fire broke out early in the morning on the balcony of a flat on the 13th floor and was contained within about two hours.
The incident happened exactly one year after the Grenfell Tower tragedy, when a fire in a 24-story residential high-rise in western London claimed the lives of over 70 people, who were trapped in their flats, as the flammable materials used for the cladding and insulation quickened the spread of the fire.