The incident happened in Arcadia Street, in the city's Bridgeton area, at about 03:00 on Tuesday.
The officer and dog were hurt as the vehicle made off. Both received treatment for non-serious injuries.
A Police Scotland spokeswoman said inquiries into the incident were ongoing and officers remained at the scene in Arcadia Street.
A police officer and police dog have been injured while responding to a report of a stolen car in Glasgow.