The driver of a black Ford Fiesta ploughed into the entrances to the Premiership side's Fair Play stand.
The turnstiles' brickwork was badly damaged in the smash, at around 5am yesterday - less than 24 hours after Celtic's victory over United on Saturday.
The driver made off from the scene and police are now working to identify the motorist. A spokeswoman for Police Scotland said anyone with information should come forward.
She added: "Anyone who might have seen the black Ford Fiesta motor car being driven in the area around that time is asked to contact Police Scotland on 101 or any police officer."