He died Monday night in Seal Beach, California, of unknown causes, the team said Tuesday.
"He was a legendary pitcher who enjoyed many of his best seasons with the Oakland A's," A's President Michael Crowley said in a statement. "He will always be a significant part of our franchise's history."
Welch was a prominent member of the Oakland teams that won three straight AL championships from 1988-90, including the 1989 club that swept the San Francisco Giants in the earthquake-interrupted World Series. He won the Cy Young Award after going 27-6 with a 2.95 ERA.
Welch finished 211-146 with 3.47 ERA in 17 seasons with the Los Angeles Dodgers (1978-87) and Athletics (1988-94). He also was the pitching coach for the Arizona Diamondbacks when they won the 2001 World Series and has served as a special instructor for the A's in recent years.
"This is a sad day for the entire A's organization," general manager Billy Beane said. "Those of us who knew Bob as a teammate and a friend will miss him greatly."
A's left-hander Sean Doolittle wrote on Twitter: "Devastated to learn of Bob Welch's passing. The A's organization lost not only one of its best pitchers, but one of its best people."