The dramatic crash in Kennington Park Road left one car overturned on the pavement, while the black cab was severely damaged.
One person, believed to be the cab driver, was taken to hospital with injuries after becoming trapped in his vehicle at about 1am.
Scotland Yard said a man was later arrested for failing to stop at the scene of an accident.
A police spokesman said: “We were called to Kennington Park Road at 12.47am to reports of a road traffic collision.
“It was reported that a person was trapped inside one of the vehicles.
“The driver of the overturned vehicle ran from the scene.
“A man was later arrested close by on suspicion of fleeing the scene of an accident. He was taken to a local police station.
“The person in the taxi was taken to hospital, but his injuries were not thought to be life-threatening”.