Sutter, who popularized the split-fingered fastball as a relief ace in the late 1970s and early 1980s, was the only player elected to the hall this year in voting by the Baseball Writers' Association of America.
He will be inducted into the Hall of Fame on July 30.
A record 520 ballots were submitted by members of the BBWAA with 10 or more consecutive years of service. Players needed to be named on 75 percent of those ballots to gain election, meaning 390 votes were required this year.
Sutter, who pitched for the Chicago Cubs, St. Louis Cardinals and Atlanta Braves, was listed on 400 ballots (76.9 percent). He was 43 votes shy of election last year when Wade Boggs and Ryne Sandberg were voted in.
Rice finished second in the balloting with 337 votes, while Gossage was next with 336. Dawson earned 317 votes and pitcher Bert Blyleven was the only other player on at least half of the ballots with 277.
The chances for these players gaining entrance to the hallowed halls next year will become slim. There were no strong first-year candidates this year, but those eligible for the first time next year include Cal Ripken Jr., Tony Gwynn and Mark McGwire.