He had musical training from the age of five; learning the violin as a pupil of Pyotr Stolyarsky and the piano with Bertha Reynbald, who also taught Emil Gilels and Tatiana Goldfarb.

Feltsman subsequently went on to collaborate with a number of prominent Russian poets including Andrey Voznesensky, Rasul Gamzatov, Eugene Dolmatovsky, Mihail Matusovsky, I. Kohanovsky, Mark Lisyansky, Lev Oshanin, Robert Rozhdestvensky, V. Kharitonov, Igor Shaferan, M. Tanich, V. Orlov, N. Olev, M. Ryabinin.

His music has been performed by notable singers including L. Utesov, Mark Bernes, Vladimir Troshin, Joseph Kobzon, Muslim Magomaev, Edita Piekha, Eduard Khil, V. Tolkunova, Yuri Gulyaev, G. Ots, Lev Leshchenko, O. Anofriev, M. Pahomenko, & Leonid Serebrennikov.

His most popular song was "Lilies of the Valley" («Ландыши»), written to verse by Olga Fadeeva (ru) and sung by Gelena Velikanova at its première.

In the mid-1980s, Feltsman arranged concerts for the instrumental ensemble The Lights of Moscow with soloist Irina Allegrova before handing over to David Tuhmanov after two successful years.