Davidovich was born in Baku, Azerbaijan SSR, into a family of musicians and began studying piano when she was six. Three years later, she was the soloist for a performance of Beethoven's Piano Concerto No. 1. In 1939, she moved to Moscow to continue her musical education. At the age of 18 she entered the Moscow Conservatory where she studied with Konstantin Igumnov and Yakov Flier. In 1949, she shared the first prize with Halina Czerny-Stefańska at the 4th International Frederick Chopin Piano Competition. This launched her on a very successful career in the Soviet Union and Eastern Europe, in which she appeared with every major Russian conductor and performed as a soloist with the Leningrad Philharmonic Orchestra for 28 consecutive seasons. She also taught at the Moscow Conservatory for sixteen years. Bella Davidovich was one of the Soviet Union's pre-eminent artists as well as one of the few women to be admitted to the inner circle of Russian cultural life. She was married to violinist Julian Sitkovetsky. Their son, Dmitry Sitkovetsky, is a violinist and conductor.
With the spirit of perestroika, she became the first Soviet émigré musician to receive an official invitation from the Soviet agency Goskoncert to perform in her native country. She played concertos, a recital with her son Dmitry Sitkovetsky playing the violin, and chamber music with the Borodin String Quartet to sold-out halls.