Mishin competed in singles within the Soviet Union and won the bronze medal at the 1964 Soviet Championships. In 1966, he took up pair skating as an experiment, teaming up with his first and only partner, Tamara Moskvina. They were coached by Igor Moskvin. Together they won the 1969 Soviet Championships, defeating both the two-time Olympic champions Ludmila Belousova and Oleg Protopopov, and the future champions Irina Rodnina and Alexei Ulanov. They went on to win silver at the 1969 World Championships. At the European Championships, they won silver in 1968 and bronze in 1969. Moskvina took time off to have a baby and they decided to retire to concentrate on their coaching careers, with Mishin focusing on coaching singles while Moskvina focused on pairs. Mishin was 28 when he retired from competition and he said he was glad to start coaching when he was young.