The family moved a lot during his early childhood. When they settled down in Moscow in 1914, he was sent to music school. He studied with Yelena Gnesin, a pupil of Ferruccio Busoni. At the same time, he studied composition with Alexander Gretchaninoff and achieved admirable results.
In 1921, Oborin was accepted into Moscow Conservatory as a student of piano and composition. He completed his piano studies in 1926. In the same year, news reached Moscow of the First International Frederick Chopin Piano Competition, to be held in Warsaw in 1927, and his piano teacher Konstantin Igumnov immediately thought of him. After winning first prize in the competition, he gave concerts in Poland and in Germany. Until 1945 he performed exclusively in Russia and taught at the Moscow Conservatory at the same time.
In 1935 he played his first concert with the violinist David Oistrakh, with whom he continued to collaborate all of his life.
Aram Khachaturian said of his Piano Concerto in D flat, "When I was working on my concerto I dreamed of hearing it played by Lev Oborin. My dream came true in the summer of 1937. The wonderful performance by this outstanding pianist ensured its success".
He gave first public performances of the works of several modern composers, including Khachaturian, Shebalin, Miaskovsky, Prokofiev (including the Violin Sonata No. 1 with Oistrakh), and Shostakovich.
He trained many pianists, including Vladimir Ashkenazy (winner of the second prize at the 1955 Chopin Competition), Mikhail Voskresensky, Dmitri Sakharov, Alexander Bakhchiev, Andrei Egorov and Boris Berman.