Isabelle Sanders, 51, died and her 85-year-old partner was seriously injured in an incident at the property in Raeswood Drive in the Crookston area of Glasgow.
Officers were called to the scene at around 1.55am today following a report of a man inside the house.
The suspect is described as white, Scottish with a local accent, aged 20-30, around 5ft 10in to 5ft 11in 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 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."
The 85-year-old man was taken to the Southern General Hospital where he is currently in a serious condition.