The crash, which occurred at about noon today (Wednesday), happened in Roseville Road.
It involved an unmarked police armed response vehicle, a parked car, another moving car and the pedestrian, a spokesman for West Yorkshire Police said.
The collision has been referred to the Independent Office for Police Conduct.
The 71-year-old woman suffered serious injuries following the crash and was taken to hospital for treatment.
Her injuries are not believed to be life-threatening.
The spokesman added: "The incident has been referred to the Independent Office for Police Conduct due to the involvement of a police vehicle and West Yorkshire Police is assisting their investigation."
Police are now trying to trace another pedestrian, who avoid the crash, and walked away from the scene.
Roseville Road is currently closed while officers are at the scene.
It is due to reopen "shortly", the spokesman said at about 5.40pm.
The witness is described as being a white man, aged in his late forties or early fifties, and was wearing a brown fur deerstalker hat, a black jacket with grey piping on the zips, light blue jeans and beige shoes. He was carrying a white carrier bag.
Anyone with information is asked to contact police via 101, quoting log number 661 of November 21.