Police pursued the blue Astra through several streets until the driver crashed it into a power box on the corner of Federal Street.
Both right hand tyres of the car were flat and the back wheel was damaged after a collision with another vehicle on Royal Street.
The driver, a woman in her 30s, and a male passenger ran from the scene.
The woman was arrested and taken to Sir Charles Gairdner Hospital for a medical check but the man has not been found.
A police spokeswoman said inquiries about the man were continuing.