Henderson, who stole 1,406 bases in a career that spanned a quarter-century, also is baseball's career leader in runs scored with 2,295. He played for nine different teams.
Rice hit 382 home runs and totaled 1,451 RBI in 16 seasons.
"I love the game of baseball," said Henderson, whose 130 stolen bases for Oakland in 1982 is the single-season major league record.
Henderson also played for the New York Yankees, Toronto, San Diego, Anaheim, the New York Mets, Seattle, Boston and the Los Angeles Dodgers.
"It is an honor to be inducted ... and have my name next to players like Jackie Robinson, Babe Ruth, Willie Mays, Hank Aaron, Lou Gehrig, Roberto Clemente and the list goes on and on," Henderson said.
Rice said, "The tears overflow because you know that the highest honor of your career means more than you ever thought it'd mean before."
Henderson was a 10-time all-star and Rice, who spent his entire career with the Boston club, was on eight all-star teams.
Also entering the Hall of Fame were Joe Gordon, chosen posthumously by the Veterans Committee; ex-Yankee Tony Kubek for broadcasting, and Nick Peters for baseball writing.