Moses started his professional career with Galveston of the Texas League, where he batted .316 in 1934. He debuted with the Philadelphia Athletics in the 1935 season. He batted over .300 each of his seven years with them, with a career-best .346 in his sophomore year.
Through years of last place finishes with Philadelphia, Moses had little chance to display his speed on the basepaths. But in 1943, with the Chicago White Sox, he posted a career-high 56 stolen bases and co-led the American League in triples (12). A strong-armed right fielder, he led the AL in putouts (329) in 1945.
In the 1946 World Series with the Boston Red Sox, Moses hit .417 (5-for-12) and tied a WS record with four hits in a game. He finished his career with the Athletics in 1951.
In a 17-season career, Moses hit .291 with 89 home runs and 679 RBI in 2012 games played. He added 1,124 runs, 2,138 hits, 435 doubles, 110 triples and 174 stolen bases. A patient hitter with a good eye, Moses collected a 1.80 walk-to-strikeout ratio (821-to-457). He also made the American League All-Star team in 1937 and 1945.