The body of Isabelle Sanders, 51, was found inside a property in Crookston, in the South Side of the city, in the early hours of Wednesday.
Police said her partner, an 85-year-old man, was injured during the incident at the property in Raeswood Drive.
Emergency services attended at around 1.55am and the pensioner was taken to the Southern General Hospital.
His condition is described as serious.
Police said the woman's death was being treated as "suspicious" and a post-mortem examination will take place to establish the exact cause of death.
Witnesses have been urged to come forward and contact police with any information.
The suspect is described as white, Scottish, with a local accent and 20-30 years old.
He is around 5ft 10in - 5ft 11in tall with a slim build and short dark hair.
At the time of the incident he was wearing jeans, a light blue top and blue canvas deck shoes.
Detective Superintendent Jim Kerr from the Major Investigation Team said: "Extensive inquiries are ongoing and specialist officers and forensic teams are currently at the scene.
"Officers are carrying out door to door inquiries and checking CCTV to assist with this investigation.
"I would ask anyone who was in the area of Raeswood Drive last night and saw a man matching the description above or anyone who recognises the description to contact police as soon as possible.
"Officers are working to establish a motive for this murder and I would like to reassure the local community that additional police patrols are in the area and anyone who has any concerns should not hesitate to speak to an officer."
Anyone with any information that may help police enquiries should contact Police Scotland on 101.