He died in a hospice in Hickory, North Carolina, after a brief illness due to complications from pneumonia, his wife Dorothy said.
Co-star John Schneider, who played Bo Duke, paid tribute to the actor calling him “a fine man”.
Dukes of Hazzard ran on CBS in the US from 1979 to 1985.
“I learned more about acting in front of a camera from Jimmie Best in an afternoon than from anyone else in a year,” Schneider told the Charlotte Observer.
His Southern twang and rugged good looks made him a natural on westerns like “Wagon Train” and “Gunsmoke” and rural series like “The Andy Griffith Show.” Among the films in which he appeared were “The Caine Mutiny” (1954), with Humphrey Bogart; “The Left-Handed Gun” (1958), with Paul Newman; “Shenandoah” (1965), with James Stewart; and “Three on a Couch” (1966), with Jerry Lewis.
His best-known role by far was Rosco on “The Dukes of Hazzard,” a countrified car-chase comedy seen on CBS from 1979 to 1985. The producers originally envisioned him as a hard-nosed sheriff, but Mr. Best saw him differently. “I said, ‘I’m going to play Rosco like a 12-year-old who likes hot pursuit,’ ” he told NPR in an interview in 2013.
In 2011 the actor sued Time Warner for using his image on Dukes of Hazzard merchandise, saying he did not receive all the royalties he was entitled.
The case was settled out of court in 2013, the day before it was due to go to trial.
Best was last seen on screen in 2013 TV movie The Sweeter Side of Life and had been due to start work on a film called Old Soldiers, about World War Two veterans.
He is survived by his wife, a son and two daughters.