The man, who has not identified himself, released his statement through the Harrisburg law firm of Andreozzi & Associates.
In response to the statement Tuesday afternoon, Clash issued a statement of his own saying he is "relieved that this painful allegation has been put to rest."
Sesame Workshop earlier this week said Clash had taken a leave of absence from Sesame Street after the allegations came to light.
In a statement, Clash admitted to having a relationship with his accuser but said it was legal and consensual.
"It was between two consenting adults and I am deeply saddened that he is trying to characterise it as something other than what it was."
Clash, a 52-year-old divorced father of a grown daughter, added, "I am a gay man. I have never been ashamed of this or tried to hide it, but felt it was a personal and private matter.
"I am taking a break from Sesame Workshop to deal with this false and defamatory allegation," he said.
Sesame Workshop said the relationship was "unrelated to the workplace."
Its investigation found the allegation of underage conduct to be unsubstantiated.
But it said Clash exercised "poor judgment" and was disciplined for violating company policy regarding Internet usage. It offered no details.
Though usually behind the scenes as Elmo's voice and animator, Clash has become a star in his own right. In 2006, he published an autobiography, My Life as a Furry Red Monster, and was the subject of the 2011 documentary Being Elmo: A Puppeteer's Journey.
Sex with a person under 17 is a felony in New York if the perpetrator is at least 21. It was unclear where the relationship took place, and there is no record of any criminal charge against Clash in the state.
Sesame Street is in production, but other puppeteers are prepared to fill in for Clash during his absence, according to a person close to the show who spoke on condition of anonymity because that person was not authorized to publicly discuss details about the show's production.
"Elmo will still be a part of the shows being produced," that person said.
In addition to his marquee role as Elmo, Clash also serves as the show's senior Muppet coordinator and Muppet captain.
He has won 23 daytime Emmy awards and one prime-time Emmy.
Clash has been a puppeteer for Sesame Street since 1984, when he was handed the fuzzy red puppet with ping-pong-ball eyes and asked to come up with a voice for him.
Clash transformed the character, which had languished as a marginal member of the Muppets family for a number of years, into a major star that rivaled Big Bird as the face of Sesame Street.
Among children and adults alike, Elmo was quickly embraced as a frolicsome child with a high-pitched giggle and a tendency to speak of himself in the third person.
"I would love to be totally like Elmo," Clash said in a 1997 interview with The Associated Press. "He is playful and direct and positive."
Besides Sesame Street, Elmo has made guest appearances on dozens of TV shows. He starred in the 1999 feature film Elmo in Grouchland.
And he has inspired a vast product line, notably the Tickle Me Elmo doll, which created a sales sensation with its introduction in 1996.
