Bluege was born in Chicago, Illinois and raised in the city's Goose Island area. He apparently did not attend high school, instead finding a job and playing baseball for local sandlot teams.
He was discovered by baseball promoter Joe Engel, who managed the Chattanooga Lookouts at Engel Stadium. He made his Major League debut on April 24, 1922 and played his final game on July 13, 1939, spending his entire 17-year playing career with the same team, then continuing with the club in managerial and executive positions.
Bluege batted .272 in his career, but was chiefly known for his defensive ability, leading American League third basemen in double plays in three different seasons. His moonlighting job earned him the nickname "The Accountant" from teammates. Bluege's only All-Star appearance, in 1935, came in a year which saw him primarily play shortstop for Washington.
After retiring in 1939, Bluege managed in the minors for four years before taking over the Senators from 1943 to 1947. He compiled a career managerial record of 379-394, with two second-place finishes. In 1948, Bluege was named the Senators' farm director, where his greatest scouting coup was a young Harmon Killebrew. In 1958 he became the team's comptroller, shortly before its relocation to Minneapolis as the Minnesota Twins. He remained the team's comptroller until his retirement in 1971, having served the organization for 50 years.
Bluege died of a stroke in 1985 in Edina, Minnesota, 10 days before his 85th birthday. The previous week, he had attended a ceremony in Washington, D.C. at halftime of a Washington Redskins football game, where Bluege was honored by being inducted in the Washington stadium's "Hall of Stars."