Four fire engines and around 25 firefighters were called to the blaze in Pownall Road at 10.38am yesterday (December 9).
A woman was among those found injured and she treated at the scene by London Ambulance Service crews.
The man and child were taken to hospital where their conditions are unclear at this stage.
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Part of the flat, located on the first floor of the building, was also damaged in the fire.
Crews from Shoreditch, Bethnal Green, Dockhead and Stoke Newington fire stations were able to get the fire under control by 11.41am.
The cause of the fire remains under investigation by the London Fire Brigade and the Metropolitan Police.
