In 1948, his friend Mario Lanza introduced him to Armando Agnini, principal stage director of the Opera in New Orleans.
As a baritone, Cosenza made his debut in New Orleans, in 1954, in a small role in Madama Butterfly, with Victoria de los Ángeles, conducted by Walter Herbert.
He went on to sing twenty-five secondary roles with the Association, most notably as Schaunard in La bohème, in 1959, opposite Licia Albanese (his favorite soprano), Giuseppe di Stefano, Audrey Schuh, Giuseppe Valdengo, and Norman Treigle, conducted by Renato Cellini, and directed by Agnini.
He also staged Il trittico, Andrea Chénier, Il barbiere di Siviglia, Carmen (with Gloria Lane, Richard Cassilly, and Treigle), Lucia di Lammermoor, La traviata (with Schuh), La bohème, Cavalleria rusticana (with Zinka Milanov), Aïda, Madama Butterfly, Don Pasquale (with Salvatore Baccaloni), Samson et Dalila (with James McCracken, Sandra Warfield, Louis Quilico, and Nicola Moscona), Rigoletto (with Roberta Peters), La traviata (with Virginia Zeani), Il segreto di Susanna, Tosca (with Gabriella Tucci, Plácido Domingo, and Cesare Bardelli), Les pêcheurs de perles, Manon (with Montserrat Caballé), Faust (with Albert Lance and Dorothy Kirsten), Macbeth (with Cornell MacNeil), Lucia di Lammermoor again (with Dame Joan Sutherland), Il trovatore, Pagliacci, Cavalleria rusticana, Manon Lescaut, Attila (with Justino Díaz), Un ballo in maschera, The Medium (with Regina Resnik), Madama Butterfly again (with Raina Kabaivanska), Lucia di Lammermoor again (with Beverly Sills), Aïda again (with Jon Vickers as Radamès), Roméo et Juliette, Rigoletto again (with Sherrill Milnes), and Andrea Chénier again (with Harry Theyard), for the company.
In 1966, his staging of the Sextet from Lucia di Lammermoor was seen on The Bell Telephone Hour's "Sights and Sounds of New Orleans," with Gianna D'Angelo, Domingo, Enzo Sordello, Thomas Paul, Bennie Ray, and Linda Neumann, conducted by Knud Andersson.