After college and the baseball draft, Stoneman pitched in the Rookie League in Caldwell, Single A in Lodi, and Double-A in Dallas in 1966. He started out the 1967 season in Double-A and moved to Triple-A in Tacoma. Stoneman was called up to the major leagues in mid-season, where he appeared in 28 games as a reliever for the Chicago Cubs, with a 3.29 ERA and 52 strikeouts in 63 innings. In the expansion draft of October 1968, Stoneman was selected by the Montreal Expos, where he spent five seasons and became a starter for manager Gene Mauch. He threw his two no-hitters with the Expos: the first against the Philadelphia Phillies at Connie Mack Stadium on April 17, 1969. It was Stoneman's fifth major league start and only the ninth game of the franchise's existence; he had eight strikeouts and five walks. The second came at the end of the 1972 season on October 2; beating the New York Mets in Montreal at Jarry Park, caught by Tim McCarver.