German-born Hague died on Monday in a hospital near Los Angeles after suffering from cancer.
As well as his role in Fame, Hague was an accomplished composer.
He wrote music for TV and Broadway shows, winning a Tony Award for Redhead in 1959.
But it was as an actor that Hague was best known and his role as the white-bearded Mr Shorofsky in the hit TV series made Hague's face familiar around the world.
Debbie Allen, who played a dance teacher in the show, paid tribute to her former co-star.
"He was such a wise, young old soul. Such a gentle, jovial person," said Allen.
Fame centred on the lives and loves of the talented pupils of New York's School for the Performing Arts.
Hague played the school's gruff but lovable music teacher for five years on television and in the 1980 film version.
Hague's other TV acting credits included guest appearances on such shows as Hotel, Beauty and the Beast and Tales From the Dark Side.
He also appeared in movies, including the Michael Jordan-Bugs Bunny comedy Space Jam, in which he played a psychiatrist.
Hague was born in Berlin but left Germany with his mother in 1937, when the Nazis came to power, to live in Rome.
He eventually settled in the US and was adopted by Dr Elliott B Hague, an eye surgeon.
Hague then studied music at the University of Cincinnati.
He was married to Renee Orin Hague, who he met while working on a musical in Cleveland in 1948.
The actress, who appeared in several of her husband's musicals, died last year.
In recent years, the couple had a successful cabaret act, appearing at Carnegie Hall two years ago. They are survived by their son Andrew.