His first major success was the first place in the Moscow Championship of 1949, ahead of players including Andor Lilienthal, Yakov Estrin and Vladimir Simagin. He became an international grandmaster in 1952. In 1954 he won the USSR Championship ahead of players including Mark Taimanov, Viktor Korchnoi, Tigran Petrosian, Efim Geller and Salo Flohr. In the 1956 Championship, he came equal first with Taimanov and Boris Spassky in the main event, finishing second after the playoff. Later Averbakh's daughter, Jane, would marry Taimanov. Averbakh's other major tournament victories included Vienna 1961 and Moscow 1962. He qualified for the 1953 Candidates' Tournament (the last stage to determine the challenger to the World Chess Champion), finishing joint tenth of the fifteen participants. He also qualified for the 1958 Interzonal tournament at Portorož, by finishing in fourth place at the 1958 USSR Championship at Riga. At Portorož, he wound up in a tie for seventh through eleventh places, half a point short of advancing to the Candidates' Tournament. He played in the 1993 Maccabiah Games in Israel, coming in fourth.