Afolabi Awodiya, 34, attacked his 20-year-old victim after she had returned to an Edinburgh flat following a night out.
The Nigerian man, who admitted to police he entered Britain illegally in 2007, had denied carrying out the rape.
But a jury at the High Court in Edinburgh yesterday found him guilty.
The woman's DNA was found on him.
Awodiya slumped forward in the dock after the verdict was announced and was led away to the cells in tears.
He told the court he had fathered two children since coming to the country.
The court heard that Awodiya knew his victim was due to be staying overnight at his former flat on February 17 last year.
She socialised with a friend and he also went out for the evening. She returned to the flat with her friend before going to sleep. Awodiya returned later.
As soon as the woman realised Awodiya was assaulting her she immediately got out of the bed and later left the flat.
Advocate depute Jane Farquharson told jurors that Awodiya appeared to have been "overcome by sexual lust" and was perhaps emboldened by drink, adding that his victim "had never shown the slightest bit of romantic interest in the accused".
She said: "He knew she was staying the night in that spare room."
Awodiya was placed him on the sex offenders register and was remanded in custody while background reports were called ahead of sentencing.