While he was never a household name, largely overshadowed by early contemporaries such as Mario del Monaco and Franco Corelli, and later ones such as Luciano Pavarotti and Plácido Domingo, Cossutta was in demand in most of the major opera houses of the United States and Europe during the 1960s and '70s. Though not the most individual or imaginative of singers, he had a stentorian, forthright tone and solid technique that served him well in the spinto tenor repertoire.
He emigrated to Argentina as a young man, where his first career was carpentry, but he turned to voice and made his opera debut as Cassio in Verdi's Otello in 1958 at the Teatro Colón. The next year saw his debut in a lead, Edgardo in Lucia di Lammermoor, also at the Teatro Colon. In 1963, he made his Chicago debut in another minor role, Abdallo in Verdi's Nabucco. Back in Buenos Aires, he came to worldwide attention when he created the demanding title role in Ginastera's Don Rodrigo in 1964, and later that year made the first of many appearances at Covent Garden as the Duke of Mantua in Verdi's Rigoletto. His Met debut was nine years later, as Pollione in Norma in 1973. In 1974, he sang the title role of Otello for the first time at Covent Garden and also made his Moscow debut as Radames in Verdi's Aida during a La Scala tour. He appeared at the Teatro Colón throughout his career, singing his last performance there as Macduff in 1998.