Rapp claims Spacey then made unwanted sexual advances toward him and grabbed his buttocks, lifted him onto a bed and laid on his body, all without the consent of Rapp, who was a minor at the time.
The 48-year-old actor then alleges he got scared and ran into the bathroom, and although Spacey tried to persuade him to stay, he left.
Rapp's lawsuit states he suffered psychological injuries, severe emotional distress, humiliation, fright, anger, depression and anxiety as a result of the alleged incident.
The documents also claim the experience has forced Rapp to step away from his work as an actor, and the actor has expressed concerns about the money he will need to spend in medical expenses to recover from the ordeal.
Rapp's lawsuit mentions another plaintiff who has remained anonymous, but who says he was also 14 and an acting student when he allegedly had intercourse and oral sex with Spacey.
The anonymous accuser claims he resisted Spacey's repeated attempts to have sex with him before he was able to flee Spacey's apartment.
The lawsuit comes three years after Rapp first accused Spacey of sexual assault, which led to the former 'House of Cards' star coming out as gay in his statement about the allegations against him.
In a statement, he said: "I have a lot of respect and admiration for Anthony Rapp as an actor. I'm beyond horrified to hear his story. I honestly do not remember the encounter, it would have been over 30 years ago."
"But if I did behave then as he describes, I owe him the sincerest apology for what would have been deeply inappropriate drunken behavior, and I am sorry for the feelings he described having carried with him all these years."
"This story has encouraged me to address other things in my life. I know that there are stories out there about me and that some have been fueled by the fact that I have been so protective of my privacy."
"As those closest to me know, in my life I have had relationships with both men and women. I have loved and had romantic encounters with men throughout my life, and I choose now to live as a gay man. I want to deal with this honestly and openly and that starts with examining my own behavior. (sic)"