By the end of the 1950s, Musheghian was one of the top featherweight wrestlers not only in the Soviet Union, but also in the world. In 1958 and 1959 he won the USSR Championships, became a bronze medalist at the 1957 World Wrestling Championships, and won the gold medal at the 1958 Wrestling World Cup. Musheghian was the first Armenian freestyle wrestler to win a World Championship medal and win the World Cup. At the USSR Second Spartakiad in 1959, he came in first place. However, in spite of these successes the USSR national team coaching staff did not select him to partake at the 1960 Summer Olympics in Rome, after which Musheghian retired from wrestling. He received the "Deserved master of sports of the USSR" title in 1966.