During his career he played for FC Dinamo Tbilisi (1965-1975), Asatiani earned 16 caps for the USSR national football team, and participated in UEFA Euro 1968 and the 1970 FIFA World Cup. At the 1970 World Cup, he was recognized as the most graceful player of the championship. He coached FC Dinamo Tbilisi between 1978 and 1982, and became the team's manager in 1987. In the 1990s, he briefly served as the chairman of the Georgian Sport Department. Subsequently, he was involved in private business and became a Vice-President of the Airzena airline. He was shot to death by unknown assailants in his own car in Tbilisi on 20 November 2002.