The 33-year-old pedestrian was struck by a grey Toyota people carrier on Queensferry Road between Clermiston and Drumbrae.
She remains in a stable condition in hospital, and it is hoped she will make a full recovery.
The incident took place at around 9.35pm on Monday.
Police are keen to trace the driver of a dark silver coupé, similar to a Toyota Celica or MR2, seen travelling northbound on Queensferry Road at the time.
A police spokesman said: “We believe that the driver of this car may hold information that can assist our inquiries, and we would urge them to contact police.”