Harrison's wife, Lori Chapman, said Tuesday that he suffered a heart attack after a performance on Saturday in Devon, southwest England, and died in a hospital.
The son of actor Rex Harrison, Noel Harrison was a British champion skier and represented the country at the 1952 and 1956 Olympics before becoming a professional musician.
He moved to the United States during the 1960s' "British invasion" and had his greatest success with "The Windmills of Your Mind," which was the theme to 1968 heist movie "The Thomas Crown Affair" and won the best-song Oscar.
Harrison wrote on his website that did not immediately realize the power of the song, by French composer Michel Legrand and American lyricists Alan and Marilyn Bergman.
"I went to the studio one afternoon and sang it and pretty much forgot about it," he wrote.
Harrison also appeared opposite Stefanie Powers in the 1966-67 TV series, "The Girl From U.N.C.L.E."
He lived in Canada for many years but returned to Britain in the 1990s. He was married three times, and is survived by his wife, several children and grandchildren.