The woman was at home in Sloy Street, in the city's Saracen area, when she became aware of a man who had entered the bedroom at about 01:00 on Friday.
Police said he subjected her to a serious sexual assault before fleeing.
The man is described as being white, about 25 years old and well-spoken with a Scottish accent. He was about 5ft 7in tall, with blond hair and was wearing black trousers.
Det Insp Craig Willison said: "This was an appalling attack which has left the woman extremely distressed. We are ensuring that the victim is being fully supported whilst we continue our inquiries.
"Finding the person who has carried out this attack is a priority for Police Scotland, and I would ask anyone who may have been in the vicinity of Sloy Street around the time of the incident to get in touch with police immediately.
''This type of attack is rare and I would like to take this opportunity to reassure the local community that we are working tirelessly to identify the perpetrator."
The detective said there would be "an increased police presence in the area" and urged anyone with information or concerns to contact officers.
A 56-year-old woman has been raped by an intruder at her home in Glasgow.