Ten fire engines and 70 firefighters scrambled to the scene on Westbridge Road at 8pm on Tuesday.
The London Fire Brigade confirmed a flat was destroyed by the fire where one woman managed to escape before crews arrived.
A woman and a child were taken to hospital by London Ambulance Service crews. A further 50 people left the building before the Brigade arrived.
Station commander Pete Johnson said: “Crews were faced with a lot of smoke issuing from the top of a block of flats on arrival.
“There was also lots of visible flame which has prompted a high number of calls to our control officers.
“Firefighters are working hard to bring the fire under control. There are a number of road closures in place.”
20th floor building on fire in battersea pic.twitter.com/uSj7nbr4Yc
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One witness told of seeing flames billowing from the top floor as fire engines arrived and people evacuated into the car park.
“The fire fighters got there quickly thank God,” she said.
“I could hear sirens for a while which isn’t normal then finally after hanging out my washing I turned the light off and saw the fire from my kitchen window.”
Officers took 18 calls to the blaze where crews from Battersea, Fulham, Wandsworth, North Kensington and surrounding fire stations are combatting the fire.
A LFB spokesperson said the brigade was called at8.02pm and the fire was under control by 8.53pm
The cause of the fire is under investigation.