The documents cite a deposition that Cosby gave in the case on Sept. 27 and 28, 2005 and quotes from it.
In the filings, which the Associated Press pushed to release, Cosby admits to giving the drug to one woman who he met in Las Vegas. It was not clear when the alleged meeting took place and the woman is not the same person who filed the suit.
"She meets me back stage," he allegedly said. "I give her Quaaludes. We then have sex."
According to the filing, Cosby's lawyer, Patrick O'Connor, said the "Cosby Show" star got the drugs in the '70s and suggested that he may have kept them for decades after that.
O'Connor would not allow Cosby to say whether he gave the drug to "other people."
The sexual abuse lawsuit was settled in 2006 for undisclosed terms, according to the AP.
Cosby has never been charged with a crime in connection to the accusations of drugging and sexual assault and his lawyers have denied the allegations.
"The only reason Mr. Cosby settled was because it would have been embarrassing in those days to put all those women on the stand and his family had no clue," the Cosby camp said in a statement to ABC News. "That would have been very hurtful."
"This admission is one that Mr. Cosby has attempted to hide from the public for many years and we are very gratified that it is now being made public," she said in a statement.
The Associated Press contributed to this report.