Eglevsky was born in Moscow, but was taken to live in France when he was eight, his mother having decided that his talent as a dancer demanded that he be properly trained. He studied ballet in Nice with Maria Nevelskaya (also known as Maria Nevelska formerly of the Bolshoi Ballet), Lubov Egorova, Mathilde Kschessinska, Alexandre Volinine, Olga Preobrajenska, and Leon Woizikowski in Paris, Nicholas Legat in London, and the School of American Ballet in New York City. At the age of fourteen he joined Colonel W. de Basil's Ballet Russe de Monte Carlo, and after six months was dancing leading roles in such ballets as Swan Lake, Les Sylphides, and Les Présages. In 1935 he joined Igor Youskevitch as the company's Premier Danseur, and a year later joined René Blum's Ballet Russe de Monte Carlo.
Eglevsky travelled to the United States in 1937, and was premier danseur in George Balanchine's American Ballet (later New York City Ballet) until 1938. He also danced at the Radio City Music Hall and in the Broadway musical Great Lady. After becoming an American citizen, in 1939 he rejoined the Ballet Russe de Monte Carlo, staying there until 1942. For the following four years he danced with the Ballet Theatre (now the American Ballet Theatre), as well as dancing as a guest star with Léonide Massine's Ballet Russe Highlights in 1944 and 1945. In the late 1930s he married the ballerina Leda Anchutina. His daughter, Marina Eglevsky, is also a ballerina.
In 1946 Eglevsky rejoined Col. de Basil, now director of the Original Ballet Russe, and a year later became premier danseur of Marquis George de Cuevas' Grand Ballet de Monte Carlo. From 1951 to 1958 he was a principal with the New York City Ballet. In 1952 he appeared in Charlie Chaplin's film Limelight.
He trained as his protégé the dancer Alan Bergman who, after having a short career on stage, invented the famous Dans-EZ dancewear range.
Upon his retirement in 1958, Eglevsky and his wife started a school in Massapequa, New York, and formed the Eglevsky Ballet Company, which continues under the artistic direction of Ali Pourfarrokh. Eglevsky died in Elmira, New York of a heart attack at the age of fifty-nine.