Cubbage originally was selected by the expansion Washington Senators in the fifth round of the 1968 MLB draft, but did not sign. He then was drafted again by Washington in the second round of the 1971 entry draft, and spent parts of four seasons in the minor leagues before joining the Texas Rangers on April 7, 1974, in a game against the Oakland Athletics. While appearing in nine games with the Rangers in 1974, he did not collect his first major league hit until being called up in 1975, in a game on June 20 against the California Angels, when he went 3-for-5 with four RBIs.