Cossette died Tuesday. The cause of his death has not been disclosed, Playbill.com said.
The lights on the theater marquees outside of the three productions of "Quartet," which Cossette helped produce, were briefly dimmed Wednesday.
"It is with great sadness that we say goodbye to our dear friend John Cossette. His passion for 'Million Dollar Quartet' was a source of constant joy for everyone involved in the show. We will miss him terribly," his fellow "Quartet" producers Gigi Pritzker, Ted Rawlins, John Gore, Beth Williams, James L. Nederlander, Joseph Smith and Michael McCabe, said in a statement.
"We are deeply saddened to report that John Cossette, 54, has passed away," the producer's family said in a separate statement issued to Playbill.com. "We ask that you please respect our privacy during this incredibly emotional time. We appreciate everyone's love and support, and further details regarding funeral arrangements will be forthcoming in the next few days."
In addition to producing "Quartet," Cossette also served as executive producer of the Grammy Awards telecast for 18 years, and the BET Awards show for 10 years.
He is survived by his wife, Rita, and daughters, Maria and Rose, Playbill.com noted.