Born in Dexter, Texas, he began his career in 1919 with the Chicago White Sox. Robertson was an average player for most of his career, having a career record of 49-80 and never winning more than he lost during a single season. His main pitch throughout his career was a slow curveball which he would often throw on the first pitch to a batter on any side of the plate, followed by a fastball up in the zone. On April 30, 1922, in just his fourth career start, he threw the fifth perfect game in baseball history against the Detroit Tigers at Navin Field (later known as Tiger Stadium) in Detroit. The Detroit lineup featured such Hall of Famers as Ty Cobb and Harry Heilmann, who both complained that he was doctoring the ball throughout the game. A spectacular diving catch by Johnny Mostil on a liner to left by Bobby Veach in the second inning preserved the historic victory. The Tigers submitted several game balls to American League President Ban Johnson after the game to check for irregularities, but Johnson dismissed the charge. No pitcher would equal the feat after Robertson for another 34 years, until Don Larsen in 1956.

After the victory, he suffered through arm troubles for the rest of his career and retired in 1928. He died in Fort Worth, Texas at age 88.