They say the man left the party at the stadium in the early hours of Sunday morning.
The 51-year old victim was seen walking outside the stadium towards Somerville Drive when a man approached her.
Detective Inspector Kate Jamieson of the National Rape Task Force said: "The woman received treatment at the Victoria Infirmary for facial injuries and was later released.
"She is extremely traumatised by what has happened and is receiving the support she needs from her family and also our partner agencies."
She added: "From studying CCTV, we believe that the suspect came out of the main entrance to Hampden Stadium just before midnight and engaged another man from the party night in conversation and it appears that the he may have provided the suspect with directions.
"The other man is then seen returning inside to the party night. Extensive enquiries are ongoing to identify the suspect, and we will trace this man.
"But I would appreciate any information from others who were at the Christmas party night who may know the man responsible, or recognise the description."
The suspect is described as white, about 30, 5ft 10 to 6ft and of medium build with short dark hair.
He has a round face and a wide smile.
He was very smartly dressed and wore a dark three-quarter length coat with distinctive buttons - possibly cream or white, a dark suit with a handkerchief in the breast pocket and a light shirt.