Bobrov started his career playing bandy hockey as a youngster. He soon was discovered by CDKA Moscow for his football talents. In 1945, he joined Dynamo Moscow as a guest player for a tour of Great Britain. He scored six goals against teams which included Chelsea, Arsenal, and Glasgow Rangers. He was highly rated by the British. During his visit in Britain, he accidentally discovered ice hockey. He literally walked into a game between two British teams and he fell in love with the sport. After returning to the Soviet Union, he started to play ice hockey for CDKA Moscow.
His successes on the ice rink exceeded that of his successes on the football field. In 1948, Bobrov scored 52 goals, compared to the runner-up who had 23. He also won the league scoring crown in 1951 (41 goals) and 1952 (34 goals). In 1950, a plane crash almost killed the entire Soviet national ice hockey team. Bobrov survived the crash as he overslept and travelled by rail. With his leadership, the Soviet Union won the world championship Gold medals in 1954 and 56. They also came second in 1955 and 57. He also won the Olympic Gold in 1956. He played 59 international games for his country and scored 89 goals.
He was enshrined into the International Ice Hockey Federation Hall of Fame in 1997, the first year it was created. For the greatest Russian athlete in the 20th century, Bobrov finished third behind football goalkeeper Lev Yashin and Greco-Roman wrestler Alexander Karelin.