Balashov was champion of Moscow in 1970 and runner-up to Anatoly Karpov in the 1976 USSR Championship.

Balashov played in four European Team Chess Championships, 1970, 1973, 1977, and 1980, winning the individual gold on the fifth board in 1977.

At the 24th Chess Olympiad in 1980, he scored 7½/10 as the first board reserve to help the USSR team win the gold medal.

According to Chessmetrics, at his peak in July 1977 Balashov's play was equivalent to a rating of 2715, and he was ranked number 11 in the world.

In 1992, he was Boris Spassky's second in Yugoslavia during the "Revenge Match of the 20th century" against Bobby Fischer.