Two fire crews from Scarborough joined police and paramedics at the scene from around 8.15pm.
Firefighters used cutting equipment to remove the roof and doors, allowing the woman to be freed.
A North Yorkshire Police spokesman said the car had been involved in a collision on the bridge heading towards the A64.
He said one person was taken to hospital by ambulance for treatment, although the severity of their injuries was not clear.
No other vehicles were involved in the incident.