Cooper signed for the St. Louis Cardinals as an amateur free agent in 1933 and played for seven of their minor league affiliates until 1938, when the Cardinals promoted him to the major leagues. After spending eight seasons with the organization, Cooper was traded in the middle of 1945 to the Boston Braves, where he spent the next two seasons. He was dealt to the New York Giants in 1947 and subsequently played for the Chicago Cubs, with whom he played his last game on May 7, 1949. He is most famous for winning the National League Most Valuable Player Award in 1942.
Cooper was born in Atherton, Missouri, and after debuting with the Cardinals in 1938, had a 12–6 record as a 1939 rookie. He was 24–21 over the next two seasons before hitting his stride, helping the team to World Series titles in both 1942 and 1944. At the start of the 1945 season, both Mort and his brother Walker staged contract holdouts, demanding that the Cardinals raise their salaries to $15,000 each. Subsequently, Mort was traded that May to the Boston Braves after only three starts; bothered by longtime elbow problems, he ended the year only 9–4. After a 13–11 season in 1946, he began 1947 at 2–5 and was traded to the New York Giants in June. He was 1–5 for the Giants over the rest of the season, and was released in July 1948 after not pitching all year due to arm trouble. He ended his career with a single 1949 relief appearance for the Chicago Cubs in which he failed to record an out. He retired with a record of 128–75, a 2.97 ERA, 913 strikeouts, and 33 shutouts in 1840⅔ innings. He was selected to the NL All-Star team four times (1942, 1943, 1945, 1946).