Scott Johnston, 19, admitted forcing his victim to the ground, pinning her down, repeatedly striking her with an iron and raping her.
The attack took place at her home in Mount Florida Glasgow on December 8, 2013, the High Court in Glasgow was told.
Johnston, a first offender, came across the woman as she walked past Hampden Stadium where he had been at a Christmas party. He followed her to her flat where he raped her.
Police released a detailed description of the attacker and, after reading it, Johnston's employer contacted them. Forensic experts found DNA matching Johnston's on the victim's clothes and on a comb he dropped.
Advocate depute Kath Harper, prosecuting, told the court the 51-year-old had been out for the evening drinking, singing karaoke and dancing with a friend in the Beechwood Bar. She left about midnight and decided to walk home.
The court heard that Johnston, a green-keeper from East Kilbride, followed his victim from Hampden Stadium.
A friend of the woman, who spotted Johnston talking to her in the street, was so worried about her that he dialled 999.
Judge Lady Rae deferred sentence on Johnston until June 25.
Defence counsel Gordon Jackson will give a plea in mitigation then.
Johntone has been in custody since December 17 last year.