Mostly known as an ice hockey player, Yevgeny Babich also excelled in football and bandy. He won the Soviet Cup in bandy in 1945-46 (as a member of CSKA Moskva) and was Soviet Champion in football in 1946 and 1947 and was runner-up in 1949 (also with CSKA Moskva). In ice hockey, Babich played with CSKA Moskva from 1944-50, then had a brief career with VVS Moskva from 1950-53, ending his career again with CSKA Moskva from 1953-57. He won eight Soviet titles (1948-53 and 1955-56). Babich was selected for the Soviet national ice hockey team from 1954-57. Besides his Olympic gold, he won World Championships in 1954 and 1956, and was second in 1955 and 1957. He also won European Championships in 1954-56 and won gold in 1953 at the World Student Winter Games.
After his playing career, Babich had a long career as an ice hockey coach. In his last active playing season (1952-53), he was a head player-coach of VVS Moskva and led his team to the Soviet title. Later he worked as a head coach of CSKA Moskva (1961-62), SKA Leningrad (1962-63), ASK Vorwärts Crimmitschau (GDR) (1964-68) and Kauchuk Omsk (1969-70). Although Babich was quite successful in his work, he suffered under clinical depression and spoke often to his friends about committing suicide, which he did in 1972 by hanging himself at his home in Moscow.