Born in Swift Current, Saskatchewan, and raised in Cold Lake, Alberta, Cleveland was listed as 6 feet 1 inch (1.85 m) tall and 195 pounds (88 kg) (13 stone, 13 lbs.).

Cleveland originally signed with the St. Louis Cardinals in 1966, and after a one-game trial with the 1969 Redbirds, he made the major leagues for good during August of 1970.

He earned a no-decision, with Red Sox relief pitcher Roger Moret gaining credit for the win when Boston prevailed, 6–3.

He also came out of the bullpen in the top of the ninth inning of Game 7 at Fenway Park and got the final out, but the Reds had already forged ahead against Cleveland's predecessor on the mound, Jim Burton, and secured a 4–3 win and the world championship.

The 1975 campaign afforded Cleveland his only postseason appearances; in his four games and 112⁄3 innings pitched, he compiled a 0–1 win–loss record and a 6.17 earned run average.