The 34-year-old pedestrian was taken to Leeds General Infirmary after being hit by a Fiat Punto on Dewsbury Road, Beeston, at around 6.37pm on Monday, May 19.
Her injuries are not believed to be life-threatening.
A West Yorkshire Police spokesman said a roads policing vehicle had tried to stop the Fiat Punto before the incident, but the car had made off.
The driver then lost control and smashed the car into several parked cars before crashing into the front of a shop near the junction of Tempest Road.
The driver, a 23-year-old man, was also injured and was later taken to hospital for treatment.
He is believed to have non-life threatening injuries.
The man has been arrested and bailed in connection with the incident.
Chief inspector Steve Palmer, of Leeds District Senior Leadership Team, said: “The area would have been busy at the time and we would like to hear from anyone who witnessed the incident or saw the circumstances leading up to it.”
Chief inspector Palmer added: “We have made the Independent Police Complaints Commission aware of the incident and have notified West Yorkshire Police Professional Standards Department, which is normal practice for an incident of this nature.”
Two fire engines from Hunslet also attended the scene on Monday.
Anyone with information about the incident is urged to contact West Yorkshire Police officers via the non-emergency number 101, quoting log number 1517 of May 19.
Alternatively, anyone with information can also call Crimestoppers anonymously on 0800 555 111.