Officers say the purple Nissan Pulsar, with the registration SE66 KNN, may have been used by the group in the attack, in the Hamiltonhill area of the city on December 16.
The car was stolen from Kilwinning on December 5 and was found burnt out on Friday on Manse Road.
Officers say a group exited the car on Salmona Street at 11.30pm on the night in question, before shots were fired at the house and the group fled in the vehicle in the direction of Auckland Street.
No one was injured in the shooting.
Officers are now looking for witnesses of the cars whereabouts between its theft and when it was found.
Detective Inspector John Morrison, of Glasgow CID, said: "We believe that the purple Nissan Pulsar may have been used by those who discharged the firearm and are appealing for anyone who may have seen the vehicle to get in touch.
"Anyone with information is asked to contact Police Scotland on 101, quoting incident 3870 of 16 January.
"Information can also be passed anonymously to Crimestoppers by calling 0800555111."