Brosius grew up in Milwaukie, Oregon, where he attended Rex Putnam High School before going to Linfield College. He was drafted by the Oakland Athletics in the 20th round of the 1987 amateur draft and signed on June 9, 1987. He became one of the limited number of players to hit a home run in his first major league game, on August 7, 1991. Brosius was the A's starting third baseman through the mid-1990s, although he played almost 300 games in his Oakland career at other positions, primarily in the outfield. In 1996, he batted .304 with 22 home runs, his best year with Oakland; however, his performace declined in 1997 (his battiong average was only .203), and he was traded to the Yankees after the season for Kenny Rogers, who had struggled mightily in New York. In his first year in the Bronx, he batted .300 with 19 home runs and 98 RBIs.

Although his performance over the next three years did not match that of his 1998 season, he remained a fan favorite. During his career with the Yankees, they won the American League pennant every year, from 1998 to 2001, as well as the World Series from 1998 to 2000. Brosius won the World Series MVP Award and was an All-Star in 1998. He won a Gold Glove in 1999.

He hit 2 home runs in Game 3 of the 1998 World Series, including one off of Padres closer Trevor Hoffman to give the Yankees a 3-0 Series lead.

He led all AL third basemen in errors in 2001, with 22, and had the lowest fielding percentage in the league (.935).

Brosius hit a two-out, two-run home run in the bottom of the ninth inning of Game 5 of the 2001 World Series against the Arizona Diamondbacks to tie the game and set up an extra-inning Yankees win. The previous night, New York first baseman Tino Martinez had hit a two-out, two-run home run to tie the game as well. It marked the first time in World Series history that this had ever occurred. The Yankees would go on to lose Games 6 and 7 of the series, after which Brosius retired.

He was given the nickname Scott Supercalifragilisticexpiali-Brosius by Chris Berman and "Brosius the Ferocious" by Yankees radio announcer John Sterling.

From 2002 to 2007, Brosius was an assistant coach at Linfield College under head baseball coach Scott Carnahan, Brosius's coach when he played for the school. In 2007, Carnahan became athletic director and Brosius was named head coach. He earned his degree from the school in 2002.

He was inducted into the Oregon Sports Hall of Fame in 2005.
In 2007, Brosius took part in the New York Yankees Old-Timers' Day festivities.