The fire broke out in the underground car park at a block of flats at the junction of Southwark Bridge Road and Great Suffolk Street.
The University College of Osteopathy clinic occupies the ground floor of the block.
Two vehicles were destroyed and four further vehicles were damaged by fire. Half of the basement was also damaged by the blaze.
Four adults and two children were treated at the scene by London Ambulance Service crews.
London Fire Brigade's Southwark borough commander Lee Sandy, who was at the scene, said: "Around 25 people left the building before the brigade arrived.
"Firefighters used fire escape hoods to rescue four people from the building. Crews wearing breathing apparatus also rescued a further two people. They were treated on scene by London Ambulance Service crews.
"Southwark Bridge Road was closed whilst crews worked to make the scene safe and local people were encouraged to find alternative routes."
The brigade's 999 control officers took 30 calls about the fire. Crews from Lambeth, Dowgate, Old Kent Road and surrounding fire stations attended the scene.
The cause of the fire is under investigation by the brigade and the Metropolitan Police Service.