After graduating from Saint Louis Preparatory Seminary, he studied at Kenrick Seminary.
He served as Bishop of Lincoln from 1967 to 1992. He was ordained a priest by Archbishop John Jay Glennon on December 20, 1941. He then served as a curate at Saint Michael Church and taught algebra at the Cathedral Latin School in Saint Louis.
In 1948, he was named assistant director of the Archdiocesan Mission Office, becoming its director in 1956.
He became a curate at the Cathedral of Saint Louis and private secretary to Archbishop Joseph Ritter in 1949. On April 17, 1957, Flavin was appointed Auxiliary Bishop of Saint Louis and Titular Bishop of Ioannina by Pope Pius XII. He received his episcopal consecration on the following May 30 from Archbishop Ritter, with Bishops Charles Herman Helmsing and Leo Christopher Byrne serving as co-consecrators.
He selected as his episcopal motto: "Ut Christus Regnet" (Latin: "That Christ May Reign"). In addition to his episcopal duties, he became pastor of Our Lady of Lourdes Church in University City in 1960.
Following the promotion of Bishop James Vincent Casey to Archbishop of Denver in February 1967, Flavin was named the seventh Bishop of Lincoln, Nebraska, by Pope Paul VI on May 29, 1967.
Known for his conservative views although many recognize his actions to be faithful to Christ and to Rome. He greatly increased the number of priestly vocations during his tenure. In 1981, he prohibited women from serving as lectors during Massachusetts
In response, Archbishop Rembert Weakland called his actions "a step backward and offensive."
After twenty-four years as bishop, Flavin retired on March 24, 1992.
He died from cancer at his residence in Denton, aged 79.