Born and raised in Detroit, Michigan, Brunett was ordained to the priesthood in Rome in 1958. From 1959–1994 he held several church positions in the Detroit area. In April 1994 he was appointed Bishop of Helena, Montana by Pope John Paul II and was consecrated bishop later that year. He was appointed Archbishop of Seattle in October 1997, and held that office until he reached his mandatory retirement age in 2010. In 2012, Brunett was appointed as apostolic administrator of the Diocese of Oakland and held that position until a permanent bishop was installed in 2013.
The second of ten surviving children, Alexander Joseph Brunett was born in Detroit, Michigan, to Raymond Henry and Cecilia Una Mary(née Gill) Brunett; His father was the first Master Plumber in the state of Michigan and completed asixth grade education at St. Charles Grade School in Detroit. His mother was born in Detroit also, but her parents were born in Ireland. They were part of the greater Irish community in Detroit at the beginning of the 20th century. His uncle and namesake was also a priest. Although four of his sisters entered the Dominican Order, three of them later left that convent to pursue other careers. As a child, Brunett sold copies of The Detroit Times. Entering Sacred Heart Major Seminary in 1946, he obtained a Bachelor of Arts degree and was valedictorian of his class. He was sent by Edward Cardinal Mooney in 1955 to study at the Pontifical Gregorian University in Rome, where he earned a Licentiate of Sacred Theology and a Bachelor of Sacred Theology. Brunett was ordained to the priesthood by Archbishop Luigi Traglia on July 13, 1958. While in Rome, he was among those selected as an honor guard for the body of Pope Pius XII after his death later that same year.
Upon his return to the United States, he served as an associate pastor at St. Rose of Lima Parish in Detroit (1959–1961) and at St. Alphonsus Parish in Dearborn (1961–1962). Brunett, while serving at St. Rose, was instructed by Archbishop John Dearden to enroll at the University of Detroit; he later received a Master of Education degree with a thesis on his experience at St. Rose entitled, "A Catholic School in a Changing Neighborhood". From 1962 to 1964, he served as a chaplain at the University of Michigan in Ann Arbor. He also became a chaplain at Eastern Michigan University in Ypsilanti in 1968. He also earned a doctorate in theology from Marquette University, and did his post-graduate studies at the Tantur Ecumenical Institute for Theological Studies in Jerusalem, the Catholic Institute in Paris, and the Goethe Institute in Germany.
Brunett taught sacramental theology and served as dean of St. John Provincial Seminary in Plymouth from 1969 to 1973, whence he was named both director of the Division of Ecumenical and Interreligious Affairs for the Archdiocese of Detroit and pastor of St. Aidan Parish in Livonia. During this time, he served as President of the National Association of Diocesan Ecumenical Officers (1974–1981) as well. Brunett co-founded and served as President of the Ecumenical Institute for Jewish-Christian Studies. He was also honored by the American Jewish Committee and received the Leo Franklin Award in Human Relations from the Temple Beth El in 1989 in "recognition of his efforts to combat anti-Semitism and to create a climate of mutual respect in Catholic-Jewish relations." Brunett was raised to the rank of Monsignor in 1990, and became pastor of the National Shrine of the Little Flower in Royal Oak in 1991. His father died two years later in 1993.
On April 19, 1994, Brunett was appointed the eighth Bishop of Helena, Montana, by Pope John Paul II. He received his episcopal consecration on the following July 6 from Archbishop William Levada, with Archbishops Elden Curtiss and Adam Maida serving as co-consecrators, in the Cathedral of Saint Helena.
Shortly after his arrival in Helena, Brunett began a series of tours of the diocese, attending welcoming ceremonies and visiting parishes. He regularly visited local Indian reservations, and was initiated into the Blackfeet tribe, receiving the name, "Holy Eagle Feather.” He was elected chairman of the United States Conference of Catholic Bishops' Committee on Ecumenical and Interreligious Affairs in 1996. His mother died that same year.
Brunett was later named the fourth Archbishop of Seattle, Washington, on October 28, 1997. Succeeding the late Thomas Murphy, he was formally installed on December 18 of that same year. He was presented with the pallium, a vestment worn by metropolitan bishops, by John Paul II on June 29, 1998.
Brunett was one of the Catholic delegates to the 1998 Lambeth Conference, and became a co-chair of the Anglican-Roman Catholic International Commission in 1999. His ecumenical activity has led him to meet such figures as Archbishop George Carey of Canterbury, Ecumenical Patriarch Bartholomew of Constantinople, and Archbishop Desmond Tutu. In 2006, in recognition of his work in ecumenical and interfaith dialogue, Brunett received an honorary doctorate from Gonzaga University.
During the Church's sexual abuse scandals, the Archdiocese of Seattle paid $31 million for settlements, counseling, and attorney fees for about 250 sex abuse victims between 1987 and 2007, but it did not file for bankruptcy, unlike the nearby dioceses of Spokane and Portland. The Survivors Network of those Abused by Priests has criticized Brunett's handling of accusations of sexual abuse against priests, despite the fact that very few accusations were made after Brunett actually became Archbishop. In regards to these cases, he has said, "It is certainly a terrible thing. It is an embarrassing thing for me personally."
Upon reaching the mandatory retirement age of 75 on 17 January 2009, Brunett submitted his letter of resignation to Pope Benedict XVI, with a request to continue in office. His resignation was accepted by Pope Benedict on Thursday, September 16, 2010. He was succeeded as Archbishop of Seattle by Bishop James Peter Sartain, who was then the Bishop of the Roman Catholic Diocese of Joliet, Illinois; he then became the Archbishop Emeritus of Seattle.
Within the United States Conference of Catholic Bishops, he currently sits on the Subcommittee on Native American Catholics and the Board of Bishops for the Pontifical North American College. He is also a member of the Board of Directors of St. Patrick Seminary and of Mundelein Seminary, as well as the Board of Trustees for the Catholic Near East Welfare Association. He is self-confessed lifelong fan of the Detroit Lions.
On September 21, 2012, the Vatican named Brunett the apostolic administrator of the Diocese of Oakland following Bishop Salvatore J. Cordileone's departure to become Archbishop of San Francisco. He served as interim ordinary for the diocese until the installation of Michael C. Barber on May 25, 2013.
On September 12, 2013, Archbishop Brunett suffered a major stroke, which put him intensive care.