He first drew international attention in 1964 when he sang the title role in the world premiere of Alberto Ginastera's Don Rodrigo, which led to a series of appearances at the Royal Opera, London during the 1960s.
Although his performance credits place him among the leading tenors of his generation, he never quite reached the star status of his early contemporaries such as Franco Corelli and Mario del Monaco, or later ones like Plácido Domingo and Luciano Pavarotti.
Notable early appearances at the Teatro Colón include Edgardo in Lucia di Lammermoor (1959), Mario Cavaradossi in Tosca (1962), and Rodolfo in La bohème (1963).
He made his American debut with the Lyric Opera of Chicago on October 4, 1963 in the role of Abdallo in Verdi's Nabucco and in that season also sang Cassio in Otello and Walter in Tannhauser.
He returned to Chicago in 1966 as Dimitri in Boris Godunov, Don Carlo in 1971, Gabriele Adorno in Simon Boccanegra in 1974, Otello in 1975, Dick Johnson in La Fanciulla del West in 1978 and Samson in 1981.
His debut at Vienna State Opera came in 1966 as Cavaradossi in Puccini's Tosca, and over the next 25 years he continued to appear there in more than 130 performances, including even the role of Froh in Richard Wagner's Rheingold.
In the 1971-1972 season, Cossutta made his debuts at La Scala, the Teatro Lirico Giuseppe Verdi, the Festival dei Due Mondi, and the Opéra National de Paris.
On February 17, 1973, Cossutta made his debut at the Metropolitan Opera in New York City singing Pollione in Bellini's Norma with Montserrat Caballé in the title part, Fiorenza Cossotto as Adalgisa, Giorgio Tozzi as Oroveso, Charles Anthony as Flavio, and conductor Carlo Felice Cillario.
In 1977, Cossutta took part in a highly lauded complete recording of Otello with Margaret Price as Desdemona, Gabriel Bacquier as Iago, and Sir Georg Solti conducting the Vienna Philharmonic Orchestra.