Taylor entered pro baseball in the Cleveland organization in 1956, attending university during off-seasons. After steady progress up the ladder in the Indians' farm system through 1961, he made Cleveland's Opening Day roster in . On April 11, the Indians' second game of the regular season, Taylor drew the starting assignment against the Boston Red Sox at Fenway Park. He proceeded to throw 11 shutout innings and collect two hits in four at bats at the plate—one half of the Indians' hit total against Boston starter Bill Monbouquette. But in the 12th inning of the scoreless tie, Taylor loaded the bases with no out, then surrendered a game-deciding, walk-off grand slam home run to Carroll Hardy and was charged with the 4–0 defeat. Appearing in eight games pitched for the Indians, he returned to the minor leagues after May 20 and spent the rest of the year in Triple-A. Then, on December 15, Cleveland traded him to the Cardinals to obtain first baseman Fred Whitfield.