Yengibaryan won a bronze medal in the bantamweight division at the 1951 Soviet Championships, and next year was included to the Soviet national team. He did not compete at the 1952 Summer Olympics due to an injury. Soviet Union debuted in the European Amateur Boxing Championships in 1953, where Yengibaryan won a gold medal in the lightweight division, becoming the first Soviet European champion in boxing. In 1954, Yengibaryan moved up to the light-welterweight division and remained at this weight. He won his first Soviet title in 1955 and would win it again in 1956 and 1958. Yengibaryan also won gold medals at the 1956 Olympics and 1957 and 1959 European Amateur Boxing Championships. He went to the 1960 Summer Olympics in Rome as a favorite, but injured a shoulder in the tournament and lost in the quarterfinals to Marian Kasprzyk.