The fire broke out in Mountacre Close near Sydenham Hill at 10.10pm yesterday (February 24).
Firefighters rescued nine people from the building who were assessed on scene by London Ambulance Service crews.
A woman and two children were taken to hospital after being treated for smoke inhalation.
Half of a storage room on the ground floor of a residential block was damaged by the blaze.
Four fire engines and around 25 firefighters from Forest Hill, West Norwood, Beckenham and Woodside fire stations attended.
The fire was under control by 12.31am.
The cause of the fire is under investigation by the brigade and the Metropolitan Police Service.
A spokesperson for London Fire Brigade said: “Residents in a first-floor flat first became aware of the fire when they heard a crackling sound coming from the rear of the block. They woke the rest of their family and called 999, while remaining in their flat due to smoke in the stairway.
“The Brigade’s Control Officers took six calls to the fire and gave vital fire survival guidance to residents inside two flats on how to stay safe until firefighters arrived.
“On arrival, crews used fire escape hoods to lead nine people, including four children, to safety.
“London firefighters were among the first in the UK to carry fire escape hoods when they were introduced in 2018 as part of improvements following the Grenfell Tower fire.
“The hoods provide members of the public with up to 15 minutes protection from four of the main fire gases (carbon monoxide, hydrogen cyanide, hydrogen chloride and acrolein) and can be worn by conscious or unconscious people.
“If more than 15 minutes protection is required then another hood can be given to each wearer. They do not provide oxygen but temporarily filter toxic smoke to make breathing easier.”
