Possessing a lyric tenor voice with a bright timbre and easy upper extension, Di Giuseppe excelled in the Italian repertory. He was particularly successful in tackeling the bel canto repertoire, notably partnering Beverly Sills in productions of Donizetti's Anna Bolena, Maria Stuarda, and Roberto Devereux, as well as Bellini's I puritani at the NYCO. Di Giuseppe performed in similar repertoire at the Met opposite other notable bel canto interpreters like Dame Joan Sutherland, Marilyn Horne and Renata Scotto.
Di Giuseppe was born in Philadelphia in 1932. His first musical experience was with the clarinet, which he played in school. He attended the prestigious Curtis Institute of Music, where he studied with Richard Bonelli. After service with the United States Army Field Band, he went on to study at The Juilliard School under Hans Heinz.
He made his formal debut in the summer of 1959, as the Chevalier des Grieux in Jules Massenet's Manon, with The Experimental Opera Theatre of America (affiliated with the New Orleans Opera Association), conducted by Renato Cellini and directed by Armando Agnini. In 1960 he made his debut with the Philadelphia Grand Opera Company as Rodolfo in Giacomo Puccini's La bohème. He sang with the PGOC several more times during the 1960s and 1970s, including portraying Pinkerton in Madama Butterfly (1963), Count Almaviva in Il barbiere di Siviglia (1963, 1975), Cassio in Otello (1964), Alfredo in La traviata (1970), and the title role in Faust (1974).
In 1962 Di Giuseppe made his debut with the San Francisco Opera as Mario Cavaradossi in Puccini's Tosca with Giulia Barrera in the title role and Chester Ludgin as Scarpia. He returned to that house two years later to sing Rodolfo to Lee Venora's Mimì. Among Di Giuseppe's other early professional opportunities were stints touring with Boris Goldovsky's New England Opera Theater and with the fledgling, and short-lived, Metropolitan Opera National Company (MONC). di Giuseppe worked for the MONC on a 1965 production of Rossini's La Cenerentola, with his wife, soprano Lorna Ceniceros. Lorna was playing the role of Clorinda in the production and Enrico was portraying Ramiro.
Di Giuseppe is one of the few opera singers to have had lengthy overlapping careers at both the New York City Opera (NYCO) and at the Metropolitan Opera. He made his NYCO debut in 1965 as Michele in Gian Carlo Menotti's The Saint of Bleecker Street. He returned in 1967 for what was possibly his greatest success, the difficult, "tenor-altino" role of the Astrologue in Rimsky-Korsakov's The Golden Cockerel, opposite Norman Treigle and Beverly Sills, conducted by Julius Rudel and directed by Tito Capobianco. He would go on to perform twenty-six roles at City Opera over sixteen years, in operas such as The Barber of Seville, The Magic Flute, Der Rosenkavalier, Cavalleria rusticana, Tosca, Manon, Gianni Schicchi, Madama Butterfly, Faust, Capriccio, La traviata, La bohème, La Cenerentola, Lucia di Lammermoor, Rigoletto, Un ballo in maschera, Roberto Devereux, Don Giovanni (directed by Frank Corsaro), Maria Stuarda, Anna Bolena, I puritani, La fille du régiment, Attila, The Makropulos Case, and Mefistofele.
Di Giuseppe's Metropolitan Opera debut took place less than five years after his NYCO debut, in a 1969 concert in Central Park as Alfredo. His first appearance on the actual stage of the Metropolitan Opera House was on January 1, 1970, opposite Martina Arroyo in Madama Butterfly. He was seen there, until 1986, in Cavalleria rusticana, The Barber of Seville, Werther (replacing Franco Corelli), Rigoletto, La fille du régiment (opposite Sutherland), La bohème, La traviata, Un ballo in maschera, La sonnambula, Faust, L'elisir d'amore, L'italiana in Algeri (with Horne), Der Rosenkavalier, Don Pasquale, Così fan tutte, Le siège de Corinthe, Adriana Lecouvreur (opposite Scotto), Samson et Dalila (as The Philistine Man, with Jon Vickers in the name part), and Rigoletto.
After the mid 1980s, Di Giuseppe was no longer appearing with either City Opera or the Met; he made his final Met appearance in 1986. In the following years, he regularly performed with the New York Grand Opera.
While working primarily in New York City, Di Giuseppe did occasionally appear as a guest artist with other opera companies. He mostly was active in the United States, singing with such companies as Baltimore Opera, the Opera Company of Boston, Cincinnati Opera, Dallas Opera, the Houston Grand Opera, and the Pittsburgh Opera among others. In 1973 he made his only appearance with the Philadelphia Lyric Opera Company as Tonio in La fille du régiment opposite Sills as Marie.
He appeared in the Opera Company of Philadelphia's (OCP) very first season in January 1976 as Pinkerton to the Cio-cio-san of Atsuko Azuma. He returned to the OCP once more in 1980 to portray Nadir in Georges Bizet's Les pêcheurs de perles. Outside of the US, Di Giuseppe made appearances with Ottawa Opera, the Canadian Opera Company, Vancouver Opera, and the Palacio de Bellas Artes in Mexico City. He also performed the role of Don José in Ireland in 1984 and made several appearances in operas in Asia from 1983 to 1985.
In the Paramount film, Foul Play (1978), a scene from The Mikado was seen. The tenor portrayed Nanki-Poo, and the scene was conducted by Rudel.
Following his retirement, Di Giuseppe taught at Florida State University and The Juilliard School. A resident of Deptford Township, New Jersey, the tenor died of cancer in Voorhees Township, on December 31, 2005. His wife Lorna Ceniceros survived him, along with a son and a brother.
He can be heard as Pollione in Bellini's Norma (1973), on ABC Records, with Sills, Shirley Verrett, Paul Plishka, and James Levine conducting his first operatic recording. In 2009, Deutsche Grammophon published it on compact disc.