Godunov was born in Sakhalin, Russian SFSR, in the Russian Far East. Godunov began his ballet studies in Riga in 1958, in the same class as Mikhail Baryshnikov. The two became friends and helped each other throughout their years there. Godunov joined the Bolshoi Ballet in 1971 and rose to become Premier danseur. His teachers there included Aleksey Yermolayev.
In 1973, he won a gold medal at the Moscow International Ballet Competition. After playing Vronsky in 1976's Anna Karenina and Lemisson, the Royal minstrel, in the 1978 film version of J. B. Priestley's 31 June, Godunov became well known in the Soviet Union as a movie actor.
On August 21, 1979, while on a tour with the Bolshoi Ballet in New York City, Godunov contacted authorities and asked for political asylum. After discovering his absence, the KGB responded by putting his wife, Lyudmila Vlasova, a soloist with the company, on a plane to Moscow, but the flight was stopped before takeoff. After three days, with involvement by President Jimmy Carter and Soviet leader Leonid Brezhnev, the U.S. State Department was satisfied that Vlasova had chosen to return to the Soviet Union of her own free will, and allowed the plane to depart. This incident was dramatised in a 1986 movie, Flight 222. Vlasova later said that while Godunov loved American culture and had long desired to live in the United States, she felt she was "too Russian" to live in the United States. The couple divorced in 1982.
Godunov joined the American Ballet Theatre and danced as a principal dancer until 1982 when he had a falling-out with long-time friend and director of the company Mikhail Baryshnikov. The official reason for his release from the company was that there would not be sufficient roles for him after a change in the repertory. He then traveled with his own troupes, danced as a guest artist with different prominent ballet companies worldwide, and turned to acting in Hollywood.
Godunov's acting roles were varied, including a good-natured Amish farmer in Witness (1985), a comically narcissistic symphony conductor (referred to as "the maestro") in The Money Pit (1986), and a violent German terrorist in Die Hard (1988). He turned down many roles which typecast him as a dancer or another heavy as in Die Hard.
Godunov married Ludmilla Vlasova, a soloist with the Bolshoi Ballet in 1971. The couple had no children and divorced in 1982.
In 1981, Godunov met actress Jacqueline Bisset at a party in New York City. They began a long term relationship six months later. They broke up in 1988.
Godunov became a naturalized citizen of the United States in 1987.
On May 18, 1995, Godunov's friends became concerned when he had been uncharacteristically quiet with his phone calls. A nurse who had not heard from him since May 8 went to his home in the Shoreham Towers, West Hollywood, California, where his body was discovered. Godunov's death was later determined to be caused by complications from hepatitis due to chronic alcoholism.
His ashes were scattered into the Pacific Ocean; his memorial at Gates Mortuary in Los Angeles is engraved with the epitaph "His future remained in the past."