The shooting happened around 8:30 p.m. Tuesday on the Beltway at I-45 South.
Investigators say the victim chased the shooting suspect to Galveston County, but haven't said whether the driver was found.
A Houston police cruiser was spotted at UTMB in Galveston after the incident.
We do not know where the victim was injured, but HPD says he or she was grazed in the shooting.