Four fire engines and around 25 firefighters were called to a fire at a shop with flats above on Upton Lane in Forest Gate.
Part of the ground floor of the building was damaged by fire. There were no reports of any injuries.
The Brigade's Fire Investigators believe the fire was accidental and involved extension leads.
A London Fire Brigade spokesperson said: "Some appliances use more power than others, so always be mindful not to overload extension leads. Power-hungry appliances, like kettles, toasters and microwaves shouldn't be on the same extension.
"It's also important to fully unwind drum extensions leads. If you leave them coiled, they can overheat and cause a fire.
"This incident also shows just how important it is to have working smoke alarms fitted on every level of your home. Smoke alarms fitted inside the property alerted two residents who were asleep on the second floor of the building, who then left the building and called 999.
“Smoke alarms give the earliest possible warning when a fire starts and we would urge everyone to make sure they have one fitted in every room where there is a risk of a fire starting, for example any room with appliances plugged in.”
The Brigade was called at 0527 and the fire was under control by 0622. Fire crews from Stratford, Plaistow, fire stations attended the scene.