Elias Grossman was born in Kobryn, Russia and emigrated to the United States and settled in Brooklyn, New York with his family in 1911. He studied at the Educational Alliance Art School; the Cooper Union; and the Art Students' League of New York. By the late 1920s, Grossman had become widely recognized for his etchings and engravings depicting New York and other American cities. In the 1930s and 1940s he traveled extensively; the etchings resulting from these travels depicted locations in southern Europe and the Holy Land.

His was a member of the Society of American Etchers, and worked for the W.P.A. Federal Arts Program. He exhibited widely during the late 1930s and 1940s. His work is held by numerous institutions including the New York Public Library, the Metropolitan Museum of Art, New York; The British Museum, and the Tel-Aviv Museum.