The thieves fled after the accident which happened around midnight at Dowan Road, Milngavie, off Auchenhowie Road.
A local resident told the Herald he was woken up by someone shouting.
He added that the car was ablaze and the fire brigade called out.
The stolen car was removed from the scene and it is believed no one was injured.
Police were only able to give the Herald the briefest of details today, adding that their enquiries were continuing.