The lawsuit stated, as reported by TheWrap: "Defendants' widely-distributed marketing campaign features a just-released trailer with explicit, profane, drug-using, misogynistic, violent, copulating, and even ejaculating puppets, along with the tagline 'NO SESAME. ALL STREET.'
"Defendants do not own, control or have any right to use the SESAME STREET mark. Instead, they are distributing a trailer that deliberately confuses consumers into mistakenly believing that Sesame is associated with, has allowed, or has even endorsed or produced the movie and tarnishes Sesame's brand."
The suit goes on to state that if viewers believe Sesame Street was behind the R-Rated movie it will cause "irreparable injury to Sesame's mark and brand."
Specifically they state that they will lose the ability to "cultivate and maintain trust with its audience of parents and young children."
"Defendants threaten to inflict serious, irreparable damage to Sesame's mark and brand by associating their adult movie with Sesame Street."
Sesame Workshop is seeking a permanent injunction preventing the use of its trademark in connection with the film's marketing. They also want unspecified damages.
The murder-mystery film in question is set in a world where humans and puppets co-exist, much like previous The Muppets movies and the show Sesame Street.
It is directed by Brian Henson, son of the late Jim Henson, who invented The Muppets.
But unlike films such as Muppet Treasure Island and The Muppet Christmas Carol, which were both also directed by Brian, this film is not affiliated with the official Muppets brand.
The Happytime Murders sees Melissa play a cop, who along with her puppet partner investigates when someone starts killing off the stars of a former hit TV show.
A spokesman for the movie production company, STX, released a statement attributed to a puppet character Fred, Esq, which said: "STX loved the idea of working closely with Brian Henson and the Jim Henson Company to tell the untold story of the active lives of Henson puppets when they're not performing in front of children.
"While we're disappointed that Sesame Street does not share in the fun, we are confident in our legal position. We look forward to introducing adult moviegoers to our adorably unapologetic characters this summer."
Along with Melissa, the film co-stars Maya Rudolph, Joel McHale, and Elizabeth Banks.