Sundberg graduated from the University of Iowa. On January 10, 1973, he was selected by Texas Rangers in the 1st round of the secondary free agent draft. In 1974, he made the rare jump from Class A level baseball to the major leagues with the Rangers. As a rookie, Sundberg was selected to be a reserve in the 1974 All-Star Game and finished fourth in the Rookie of the Year balloting (teammate Mike Hargrove won the award).Sundberg had 101 assists in 1975, becoming the first American League catcher to have more than 100 assists in a season since the end of the Second World War.

Sundberg established himself as one of the top defensive catchers in the American League by winning six consecutive Gold Glove Awards from 1976 to 1981. In December 1983, after ten years with the Rangers, he was traded to the Milwaukee Brewers. After playing one season with the Brewers, he was traded to the Kansas City Royals.

Sundberg's veteran experience helped bolster the Royals' young pitching staff, and the team's combined earned run average improved to second best in the American League. The Royals went on to win the 1985 World Series. In Game Six of that series, Sundberg scored the dramatic ninth-inning winning run by sliding into home plate, skillfully avoiding the tag of St. Louis Cardinals catcher Darrell Porter. Sundberg reached base when he laid down a sacrifice bunt that resulted in a force out at third.

In 1986, Sundberg helped the Royals pitching staff lead the league in earned run average, however they fell to third place in the American League's Western Division. Sundberg was traded to the Chicago Cubs in 1987, before signing back with Texas where he ended his career at the end of the 1989 season.