His family told The New York Times that Odom suffered a fatal heart attack Friday at his vacation house in Lincoln, Vt.
Odom worked in the NSA under President Jimmy Carter and headed the agency under President Ronald Reagan. During a long career as an Army officer and in government, he opposed detente with the Soviet Union.
Zbigniew Brzezinski, who first met Odom when he taught him at Columbia University, said he was surprised when the Vietnam veteran argued that the war helped Russia by containing China.
Odom opposed the Iraq War before the invasion, arguing that the country would be easy to conquer and hard to hold. In May 2006, he published an article in Foreign Policy titled "Cut and Run? You Bet."
The son of an agricultural researcher in Tennessee, Odom graduated from West Point in 1954. He is survived by his wife, Anne Weld Curtis, curator emeritus of the Hillwood Estate Museum and Gardens in Washington, and a son, Lt. Col. Mark Odom, who was wounded in Iraq last year.