Though he left Russia as a child during the Revolution, Eglevsky acquired the traditional style and technique of the Imperial Russian Ballet by studying in Paris (from age eight) with such outstanding émigré dancers as Lubov Egorova, Mathilde Kschessinska, and Alexandre Volinine and with Nicholas Legat in London. At 14 he became premier danseur with Colonel W. de Basil's Ballet Russe de Monte Carlo, in which he introduced his characteristic and spectacular series of slow, controlled pirouettes in Les Présages.

He danced with such companies as the René Blum–Michel Fokine Ballets de Monte Carlo, the American Ballet, and the Ballet (now American Ballet) Theatre before joining the New York City Ballet (1951–58), where he created leading roles in several George Balanchine ballets, including Scotch Symphony (1952) and Caracole (1952; now called Divertimento No. 15). Among his other well-known roles were Albrecht in Giselle and Prince Siegfried in Swan Lake; principal parts in Fokine's ballets Les Sylphides, Prince Igor, Le Spectre de la Rose, and Petrushka; the burlesque Paris in David Lichine's Helen of Troy; and the title role of Léonide Massine's surrealistic Mad Tristan. Eglevsky, a U.S. citizen from 1937, established a school and small performing group in Massapequa, N.Y., in 1958, which grew into the Eglevsky Ballet Company that survived him.