His father was German Jewish,[3] and his ancestors were named Auerbach, meaning "meadow brook".

Averbakh's first major success was taking 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 by qualifying for the 1953 Candidates' Tournament (the last stage to determine the challenger to the World Chess Champion), finishing joint tenth of the fifteen participants.

In 1954, Averbakh won the USSR Championship ahead of players including Mark Taimanov, Viktor Korchnoi, Tigran Petrosian, Efim Geller and Salo Flohr.

He also qualified for the 1958 Interzonal tournament at Portorož, by finishing in fourth place at the 1958 USSR Championship at Riga.

At Portorož, Averbakh wound up in a tie for seventh through eleventh places, half a point short of advancing to the Candidates' Tournament.

From 1956 to 1962 he edited (with Vitaly Chekhover and others) a four-volume anthology on the endgame, Shakhmatnye okonchaniya (revised in 1980–84 and translated as Comprehensive Chess Endings, in five volumes).