Sundberg graduated from the University of Iowa. While attending the University of Iowa he joined The Delta Upsilon Fraternity. On January 10, 1973, he was selected by Texas Rangers in the 1st round of the secondary free agentdraft. 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.
In a sixteen-year major league career, Sundberg played in 1,962 games, accumulating 1,493 hits in 6,021 at bats for a .248 career batting average along with 95 home runs, 624 runs batted in and an on-base percentage of .327. His .993 career fielding average was 8 points higher than the league average over the span of his playing career. Sundberg led American League catchers six times in fielding percentage, putouts and assists. He completed 145 double plays in 1962 games in his career, and holds the major league record for the best ratio of double plays to errors of any catcher in major league history behind the plate for at least 1000 games.
Sundberg was the first catcher to win six American League Gold Gloves, although Bob Boone won five in the American League and two more in the National League. His 1976 Gold Glove was the first by any Rangers player. He caught 130 shutouts in his career, ranking him fifth all-time among catchers. Sundberg played more games as a catcher than any other player in Rangers history (1512). At the time of his retirement, Sundberg had caught more major league games than any man in history except his contemporary Bob Boone. He still ranks fifth today. Richard Kendall of the Society for American Baseball Research devised an unscientific study that ranked Sundberg as the third most dominating fielding catcher in major league history.
Galesburg High School has named their main baseball field after Sundberg. Sundberg stepped down from his position as senior executive vice president with the Rangers on July 2, 2014, after spending 10 years in the front office.