The club have also admitted liability for historic abuse carried out by McCafferty while he worked at the club as a coach.
The award was made to the victim, who cannot be named, by the Personal Injury Court in Edinburgh.
The victim’s lawyer, Kim Leslie, said: “For decades my client suffered as a result of McCafferty’s abuse so above all else I’d like pay tribute to him for the courage he has shown in stepping forwards after coming to terms with what happened.
“Ever since McCafferty’s abuse was exposed Celtic FC remained firm in its defence and denial of responsibility.
“However after thorough investigations I was able to present a robust case which resulted in the club admitting liability.
“I hope the successful end to this legal action brings closure to my client and also brings hope to those who considering a civil action of their own.”
Although a compensation figure has not been disclosed, it has been described as “significant”.
McCafferty was jailed for six years and nine months in May after admitting to 12 charges related to child sex abuse against 10 teenage boys between 1972 and 1996. The 73-year-old was a coach and kit man for the Celtic youth team, who also worked for Celtic Boys Club.