Bezsonov's career began in 1976 with Dynamo Kyiv. Here he spent most of his career, except for a short one-season stint in Israeli club Maccabi Haifa F.C. in 1990–91. He was well known as a lightning quick full-back with a powerful shot. Bezsonov was attack- minded and scored a good return for a defender. He ended 27 goals in the 377 games that played for Dynamo.
Bezsonov won 79 caps and scored 4 goals for the USSR national football team from 1977 to 1990 and was included in three FIFA World Cup squads. He also holds the record for the fifth highest appearances for the USSR national football team. In the 1990 World Cup he was sent off against Argentina. His team ended up losing the game 2–0 which became eventually his farewell match for the Sbornaya.
In 1979 Bezsonov played couple of games for Ukraine at the Spartakiad of the Peoples of the USSR.
Bezsonov is married to Viktoria Serkyh, a former two-time world champion in rhythmic gymnastics. Together they have a daughter, Anna Bessonova, who also competed internationally in rhythmic gymnastics, became world champion in 2007, and an Olympic bronze medalist in 2004 and 2008.