Finan graduated from the University of Maryland School of Law in 1939 and was admitted to the bar the same year.
He served from June 1941 to September 1945, which included 23 months in the European theater with the 45th infantry division.
During his tenure as solicitor, he was responsible for cases relating to the Cumberland-Ridgeley flood protection project.
He was appointed as attorney general of Maryland by Governor J. Millard Tawes in 1961, succeeding C. Ferdinand Sybert.
While attorney general, he presided over the U.S. Supreme Court case of Madalyn Murray O'Hair to ban prayer in public schools.
[1] He served as president of the lay board of the Sacred Heart Hospital in Cumberland.
[2] Finan received an honorary doctorate degree from Mount St. Mary's University in 1965.