Welch won the AL Cy Young Award after going 27-6 with a 2.95 ERA in 1990. His 27 wins tied him with Steve Carlton in 1972 for the most in a single season since Denny McLain's 31 victories in 1968.
"He was one of the greatest competitors to wear the Dodger uniform," Dodgers president and CEO Stan Kasten said.
Several current A's players offered their condolences on Twitter. A's left-handed reliever Sean Doolittle took to Twitter on Tuesday, saying he was "devastated" by Welch's death.
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. #RIP35 - Sean Doolittle (@whatwouldDOOdo) June 10, 2014
No one answered the door Tuesday at Welch's home, which had been sealed with a sticker from the Orange County coroner.
Neighbor Alma Purcha said she woke up to find police cars outside the home several blocks from the Pacific Ocean. She said Welch divided his time between Arizona and Seal Beach. She last saw him with his son and daughter Friday, when they exchanged pleasantries.
The A's said Welch is survived by his sons Dylan, 25, and Riley, 23, daughter Kelly, 18, and former wife Mary Ellen.
ESPNLosAngeles.com's Mark Saxon and The Associated Press contributed to this report.