Born in Jessup, Pennsylvania, Cossa studied with Anthony Marlowe in Detroit, Michigan, Robert Weede in Concord, California, and Armen Boyajian in New York City. He made his debut at the New York City Opera as Moralès in 1961, and a week later sang Sharpless with the company. He won the American Opera Auditions in 1964 and was sent to Italy for debuts at the Teatro Nuovo in Milan and Teatro della Pergola in Florence.
He made his debut at the San Francisco Opera in 1967 as Zurga in Les pêcheurs de perles. His Metropolitan Opera debut took place on January 30, 1970 as Silvio in Pagliacci. Other roles there were Figaro in Il barbiere di Siviglia, Lescaut in Manon Lescaut, Marcello in La bohème, Mercutio in Roméo et Juliette, Masetto in Don Giovanni, Valentin in Faust, Yeletsky in Pique Dame, Germont in La traviata, and Albert in Werther. In 1976 he created the role of David Murphy in the world premiere of Gian Carlo Menotti's The Hero with the Opera Company of Philadelphia.
Cossa's left a few notable recordings of his best roles such as Belcore in L'elisir d'amore opposite Dame Joan Sutherland and Luciano Pavarotti, Achillas in Handel's Giulio Cesare opposite Norman Treigle and Beverly Sills, Nevers in Meyerbeer's Les Huguenots, again opposite Sutherland, Martina Arroyo and Huguette Tourangeau, and the baritone solo part in Roger Sessions' When Lilacs Last in the Dooryard Bloom'd. He can also be heard on the Classical Record Library's A Celebration of Schumann and Schubert with the Chamber Music Society of Lincoln Center.
He was chosen by Licia Albanese to be the recipient of the Puccini Foundation's Bacccarat Award in 2004, and in 1993 was inducted into the Hall of Fame for Great American Singers at the Academy of Vocal Arts in Philadelphia.
- Bender, William. "Music: Souvenir Opera", Time (June 14, 1976)
- Central Opera Service Bulletin. "Winners of Vocal Competitions" (May–June 1966)
- Hamilton, David (ed.) "Cossa, Dominic", The Metropolitan Opera Encyclopedia. Simon and Schuster (1987) ISBN 0-671-61732-X
- The New York Times, "4 American Opera Singers Have Debut in Milan" (September 21, 1962)
- University of Maryland School of Music. Faculty biography: Dominic Cossa
- Opera Music Theater International. Dominic Cossa: Opera Music Theater International Lifetime Achievement Award
- Cummings, David (ed.). "Cossa, Dominic", International Who's Who in Classical Music 2003. Routledge (2003). ISBN 1-85743-174-X