Police were called to Ashton Terrace, Harehills, at the junction with Ashton Road, following the crash at around 8.15pm last night (Friday).
Both the woman, 37, and the girl were knocked to the ground and taken to hospital with minor injuries after the collision.
Police said the driver of the car, which was red, did not stop.
PC David Webster, of Safer Roads and Neighbourhood Support East, said: "We are carrying out extensive enquiries to trace the red car and the driver involved and we would like to hear from anyone who witnessed the collision, saw the car in the area or who has any information that could assist the investigation."ã€€
Anyone with information is asked to contact PC 5979 Webster at Carr Gate via 101 quoting log number 1558 of January 20 or [email protected]
People can also call the independent charity Crimestoppers anonymously on 0800 555 111.