Originally a football player, he played for CDKA Moscow, VVS Moscow, and Spartak Moscow, and represented the Soviet Union internationally at the 1952 Summer Olympics. After he quit football in 1953 he turned to ice hockey, which he had taken up when it was started in the Soviet Union in 1946. He was one of the first ice hockey players in the Soviet Union, and joined CDKA Moscow, playing for them and VVS Moscow before retiring in 1957. A leading scorer in the Soviet League, Bobrov was one of three players to average more than two goals per game over their career, with the other two players (Alexei Guryshev and Viktor Shuvalov) his linemates. Internationally he participated with the Soviet national team at several World Championships, including their first tournament in 1954, as well as the 1956 Winter Olympics, where the Soviets won the gold medal.