Flanagan lettered for the UMass baseball team in 1972 and 1973, earning first team All-Yankee Conference and first team All-New England honors in 1973 before turning professional. Flanagan went 9-1 with a 1.52 ERA and 91 strikeouts in 1973, to lead the team in all three categories. His nine wins and .900 winning percentage set school single season records at the time. He received the E. Joseph Thompson Memorial Trophy as the team's MVP in 1973, after leading the team to an overall record of 21-9-1, including a trip to the NCAA Tournament and the Yankee Conference championship.
Flanagan had a career ERA of 1.19 and a career winning percentage of .923 (12-1), which are both still the best marks in school history. He also played in the outfield while at UMass, hitting .320 with six homers and 29 RBIs in 128 career at-bats.
After he was drafted by the Baltimore Orioles in 1973, Flanagan went on to pitch 18 seasons in the major leagues with the Orioles (1975-1987, 1991-1992) and Toronto Blue Jays (1987-1990). He had a career record of 167-143 with 1,491 strikeouts and a 3.89 ERA in 2,770 innings pitched, and won the 1979 American League Cy Young Award for the Orioles, going 23-9 with 190 strikeouts and a 3.08 ERA in 265-2/3 innings. Flanagan pitched over 200 innings seven times in his major league career, including four straight years from 1977-1980. He won 12 or more games in a season eight times during his major league career, struck out 100 or more batters six times, and was a member of the Baltimore Orioles 1983 World Championship team.
A native of Manchester, N.H., Flanagan received his degree from the UMass School of Education in 1975, and was inducted into the Hall of Fame in 2000.
Since 2010, Flanagan had served as a color analyst on Orioles telecasts for Mid-Atlantic Sports Network.