Azito was part of Juilliard's famous "Group I," the first students admitted to the drama program administered by John Houseman. His fellow students included Patti LuPone and Kevin Kline. Soon after arriving, Azito fell under the influence of choreographer Anna Sokolow and began studying modern dance — although, at six-foot-three (190 cm), Azito was an unusual candidate for dance training. (There was another dancer in the family: Azito's younger brother, Arturo Azito, performed with Eliot Feld and the Boston Ballet.) This newfound interest in dance aggravated Houseman, who was apparently anxious about the number of gay men in Group I and had already clashed with Azito over a cross-dressing incident. Partly as a result of his conflict with Houseman, Azito left Juilliard without taking a degree and, as "Antonio Azito," spent two years performing in Sokolow's company.
Returning to drama in the mid-1970s, Azito began working in avant-garde off- and off-off-Broadway theater, including Cotton Club Gala, Bebop, The Life and Times of Toulouse Lautrec, and C.O.R.F.A.X. He quickly became associated with the director Wilford Leach, who would be one of Azito's most frequent employers until Leach's own death. He made his Broadway debut in Richard Foreman's controversial revival of The Threepenny Opera, in a dancing role ("Samuel") invented just for him. Critics were intrigued by what soon became known as Azito's signature: a dancing style that made him look like a somewhat off-kilter marionette, accompanied by stylized facial expressions. An interviewer once described him as "a bit like Buster Keaton injected with Silly Putty." This production also inaugurated Azito's association with Joseph Papp's New York Shakespeare Festival, which continued with another Brecht-Weill musical, Happy End (1977).
Azito's best-known role, however, came in yet a third production for NYSF: as the Sergeant of Police in the 1980 Broadway revival of The Pirates of Penzance, starring Linda Ronstadt and Kevin Kline. His performance earned him a Tony Award nomination and a Drama Desk Award, and he repeated the role in the 1983 film version. Azito went on to perform at Radio City Music Hall, the Mark Taper Forum, and in the abortive American National Theater company at Kennedy Center. After playing Feste in the NYSF production of Twelfth Night (1986), directed by Wilford Leach, Azito's career in New York City virtually ceased; his last Broadway role was in The Mystery of Edwin Drood, also directed by Leach.
Azito's one film role of note was as the Sergeant in The Pirates of Penzance. He played bit parts in a few films, most notably Woody Allen's Stardust Memories (1980) and Moonstruck (1987), and was the lead in the cult film Apple Pie (1976). Azito also has a cameo as one of the party dancers in The Addams Family (1991). On television, he played guest roles on Miami Vice, The Equalizer, and Beacon Hill.
Azito continued working in regional theater and occasional films until 1994, approximately a year before his death from AIDS.