At the High Court in Glasgow, 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.
Johnston, a first offender, came across the woman as she walked past Hampden Stadium, where he had been attending a Christmas party. He then 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 had dropped.
Advocate depute Kath Harper, prosecuting, told the court that the 51-year-old had been out for the evening drinking, singing karaoke and dancing with a friend in the Beechwood Bar and left there about midnight and decided to walk home.
Ms Harper said: "She does not remember leaving the pub or how she got home."
The court heard that Johnston, a greenkeeper from East Kilbride, followed his victim from Hampden.
Ms Harper added: "After being in the Beechwood Bar the next thing the woman remembers is being in her living room lying face down on the floor.
"Her head was being held down by someone behind her and she could not move."
The 51-year-old reached out for an iron which was on the floor next to her and swung it, and believes she hit Johnston with it.
Judge Lady Rae deferred sentence on Johnston until June 25 at the High Court in Glasgow. He has been in custody since December 17 last year.
Defence counsel Gordon Jackson will give a plea in mitigation then.