Godunov was the latest in a string of Soviet ballet dancers to defect to the United States in the 1960s and 1970s. Rudolf Nureyev (1961), Natalia Makarova (1970), and Mikhail Baryshnikov (1974) had all sought and received asylum in the United States and went on to pursue successful dance careers in America and around the world. Godunov was in a different class, however. A New York Times article discussing his defection stated that "with his mane of long blond hair and powerful tall build, Mr. Godunov may well be the premier danseur of the rock generation. ...Young audiences identify with him totally." That may have been the problem for Godunov. Beginning in 1974, he was banned from performing anywhere outside of Russia for four years because his "hippie-like demeanor both on and offstage may have been too flamboyant." Godunov's defection in August 1979 was also noteworthy because he was the first dancer from the prestigious Bolshoi Ballet to seek asylum in America. The previous defectors had been from the Kiev opera company. For decades, the Bolshoi had been Russia's cultural jewel, touring the world numerous times. Godunov's departure was a serious blow. His wife, Ludmila Vlasova, did not join her husband in defecting and returned with the Bolshoi Company to Russia.
For the next three years, Godunov danced with some of the most impressive dance companies in America, including the American Ballet Theater and the Alvin Ailey Dance Theater. He also pursued an acting career in Hollywood, appearing in many films, including Die Hard and Witness. Godunov died in 1995.