Greg Moffat, 22, from Airdrie, attacked the 23-year-old woman at a flat in the North Lanarkshire town in the early hours of 16 March 2014.
The High Court in Glasgow heard he told his victim he was sorry but later claimed to police that they had been involved in a consensual sex act.
Judge Lord Armstrong placed Moffat on the sex offenders register, remanded him in custody and deferred sentence.
The court heard that the victim was part of a group who had gone back to the flat following a night out at a local club.
She ended up going to sleep in a room in the flat but was later woken by Moffat raping her.
The woman told the jury: "I said to him: 'You were having sex with me when I was asleep. I said it over and over'.
"I was shocked. I didn't know what to think - can't explain it. I just wanted to get away."
Moffat later apologised to the woman but after being arrested he insisted they were involved in a consensual sex act rather than full intercourse.
However, the woman told the court she wanted her attacker to tell the truth.
She said: "I don't know why he is just not admitting it. I cannot believe why he would say that. I don't know why this happened."
The jury was also told that the rape had changed the woman's life and she had given up her job.
She added: "My whole life has changed. I don't feel comfortable with people. I am just different. I just don't have the same life that I did."
Following the guilty verdict, first offender Moffat wept as he was led handcuffed to the cells.
A man has been convicted of raping a woman while she was asleep.