The Metropolitan Police said the Ford Mondeo had failed to stop for officers and an authorised pursuit was under way by a marked police car.
The Mondeo mounted the kerb and drove on the pavement on Tottenham High Road, shortly before 15:50 BST.
It hit a woman in her 50s and an eight-year-old boy and did not stop.
The boy's injuries are not thought to be life-threatening.
Police said they were waiting for an update on the condition of the woman.
Officers said the car carried on driving after hitting the pedestrians and went back on the road, before it hit a wall in Park Lane.
The occupants then abandoned the vehicle and fled.
Officers who were in the police car stopped and went back to help the pedestrians until London Ambulance Service arrived.
Collision investigation officers remain at the scene.
The police force said there had been no arrests and inquiries continued.
A woman and a child have been taken to hospital after they were hit by a car that mounted a kerb near White Hart Lane stadium during a police chase.