The son of Charles Denby, Jr. and Martha Dalzell Orr, Edwin was born in Tientsin, China in 1903, where Charles had been appointed as Chief foreign advisor to Yuan Shikai a year earlier. Edwin's grandfather, Charles Harvey Denby, who had served as United States Ambassador to China for an unprecedented 13 years, died when Edwin was one. Edwin spent his childhood first in Shanghai, then in Vienna, where his father served as consul general from 1909-1915, before coming to the United States in 1916. He was educated at Hotchkiss, and attended Harvard but failed to graduate. He also attended classes at the University of Vienna, before obtaining a diploma in gymnastics (with specialty in modern dance) at the Hellerau-Laxenburg school in Vienna in 1928. He performed for several years, notably with the State Theater of Darmstadt.

Looking for someone to take his passport photo, he encountered photographer and filmmaker Rudy Burckhardt in Switzerland in 1934, and the two remained inseparable for the rest of Denby's life. The following year, they returned to New York and rented a loft for eighteen dollars a month in a five-story walk-up building on West 21st Street in Chelsea. Edwin's friendship with painter Willem de Kooning, who lived one floor below in the adjacent building, began shortly thereafter when de Kooning's kitten turned up on the fire-escape outside of Edwin's window one evening.On Tuesday, July 12, 1983, at the summer house he maintained with Burckhardt in Searsmont Maine, he committed suicide; he had been ill and increasingly concerned about the loss of his mental powers.

Denby was inducted into the National Museum of Dance C.V. Whitney Hall of Fame in 2002.

In 1935, soon after Denby's return to New York, Orson Welles and John Houseman asked him to adapt Eugène Labiche's Un Chapeau de Paille d'Italie for the stage. The play, titled Horse Eats Hat was scored by Paul Bowles and was performed as a Works Progress Administration Federal Theatre Production in 1936.During his lifetime, being ambivalent about the publication of his poetry, he was known primarily as a dance critic. At the behest of Aaron Copland and Virgil Thomson, he began writing a dance column for the magazine Modern Music in 1936. In 1943, Thomson drafted Denby as the dance critic for the Herald Tribune.