Moore was accused of sexual misconduct during his campaign for the Alabama Senate previous year. He has denied the misconduct allegations.
The Moores are being represented by Larry Klayman, founder of Judicial Watch and Freedom Watch.
"As a direct and proximate result of Defendant Cohen's and his agents' including Defendants Showtime's and CBS's false and fraudulent representations, Judge Moore, Mrs. Moore, and his entire family have suffered extreme emotional distress as a result of Judge Moore being falsely portrayed as a sex offender and pedophile in this district, on national television and worldwide".
Former Alabama senate candidate Roy Moore appeared on 'Who Is America?' in July on the pretence of accepting an award for his support for Israel.
It says Cohen's interview "falsely painted, portrayed, mocked and with malice defamed Judge Moore as a sex offender".
According to the lawsuit, Cohen, "while in character, falsely and fraudulently induces unsuspecting victims, such as Judge Moore to be interviewed under dishonest, unethical, illegal and false pretenses".
The show begins as Cohen, in full transformative makeup, discusses the importance of technology in the military.More news: White House: Iran, Russia Gear Up for Mass Strike in Syria
In the show broadcast in the United States in July, when the beep sounds, Mr Moore tells Baron Cohen: "I support Israel".
When the device beeped, Mr Moore stopped the interview and said: "I've been married for 33 years".
Moore is suing for damages of $95 million and has asked for as a jury trial.
Showtime issued the following statement: "The press has been sent copies of an alleged complaint, yet to our knowledge SHOWTIME has not been served".
Representatives of Baron Cohen did not immediately respond to requests for comment about the lawsuit.
Moore's lawyer wrote that as they trying to stop the segment from airing, "defendants CBS, Showtime and thus Cohen were informed that the release that Judge Moore had signed was obtained through fraud, and was therefore void and inoperative".