Half of a four-bedroom flat was damaged in the fire in Wards Wharf Approach in Silvertown around 2.53pm yesterday (February 24).
Half a balcony was also damaged while another balcony was destroyed.
The woman and baby left the property before the brigade arrived and there were no injuries.
The fire service believes the fire was accidentally started by “unsafe disposal of smoking materials”.
Four fire engines and around 25 firefighters from Plaistow, Poplar, Stratford and Millwall fire stations put the fire out by 3.40pm.
A London Fire Brigade spokesperson said: “If you’re a smoker, it’s absolutely vital you ensure your cigarette is completely out when you’ve finished smoking it.
“If you don’t, you risk causing a fire which could not only destroy your home, but also cost you your life.
“Firefighters recommend that you have a look at our online Home Fire Safety Checker to make sure there are no hidden hazards in your home.
“The online tool will ask you a series of questions that will help us work out the level of risk in your home, or the home of someone you care for.
“It only takes a few minutes to get tailored advice to keep yourself and loved ones safe from fire.”
