Following the death of Bishop Edward Kelly, Byrne was appointed the fourth Bishop of Boise, Idaho, on June 16, 1956. His installation took place at Cathedral of St. John the Evangelist on August 29 of that year. He remained in Boise for nearly six years.
Following the appointment of Archbishop Leo Binz to the Archdiocese of Saint Paul, Pope John XXIII named Byrne the ninth bishop and seventh archbishop of Dubuque on March 7, 1962. He was installed as archbishop by the Apostolic Delegate on May 8, 1962, in St. Raphael's Cathedral. Archbishop Byrne would serve as the leader of the Dubuque Archdiocese for 21 years. From 1962-1965, Byrne was present at all four sessions of the Second Vatican Council. Following the council he was responsible for implementing the reforms promulgated by the council within the archdiocese. During his time in office, Archbishop Byrne maintained a relatively low profile in the community. He was noted for asking that Catholics join him in a televised recitation of the Rosary before the evening news, and had cards produced that people could place on top of their television sets. Archbishop Byrne also encouraged the growing cooperation among the Roman Catholic and Protestant seminaries and colleges in Dubuque and often accepted invitations to participate in seminars and to speak to classes. On one occasion, in an airport, he encountered a Protestant seminarian he had come to know through these activities who was going to interview for a parish position and offered his blessing for his success and for his future ministry. He also devoted as much time as he could to visiting those who were hospitalized in Dubuque. In 1966 the archdiocese was one the nation's first dioceses to establish a Priests' Senate. The following year the Archdiocesan Board of Education was established, and adult education was started at the parish level. The Priest Personnel Board was established in 1968 to advise the archbishop on priest assignments in the archdiocese. In 1970 an Interim Pastoral Council was developed as an advisory and consultative group. Villa Raphael was opened in 1971 as a home for retired priests. On October 4, 1979, Pope John Paul II made his historic visit to Iowa with Archbishop Byrne serving as his official host in his role as "Primate of Iowa." Byrne was also Chancellor of Loras College. Pope John Paul II accepted Byrne's resignation on August 23, 1983, and he was named Archbishop Emeritus of Dubuque. Following his retirement, Byrne remained in the Dubuque area. He died while a resident at the Stonehill Care Center in Dubuque on August 2, 1996, and was buried in the mortuary chapel of St. Raphael's Cathedral along with other deceased bishops and archbishops of the archdiocese. Archbishop Byrne had earlier purchased a simple wooden casket from the monks of New Melleray Abbey and stored it at the Cathedral; this was the casket in which he was buried.