Esposito was Presley's professional assistant and a member of his inner circle, known as the "Memphis Mafia".
He was one of two best men at the star's wedding to Priscilla in 1967 and a pallbearer at his funeral in 1977.
Esposito's daughter Cindy Bahr said he had died of natural causes in California on 23 November after suffering from dementia.
The pair met while serving in the Army in 1959. When they completed their service, Esposito became Presley's assistant, a role that saw him acting as road manager when Elvis was on tour.
He was a member of the so-called Memphis Mafia with Red West and Marty Lacker, and appeared in some of Presley's films, including It Happened at the World's Fair.
Esposito was one of those who found Presley's body when he died at Graceland, Memphis, and broke the news of his death to Priscilla and Presley's manager Tom Parker.
He went on to work as a road manager for artists including Michael Jackson and The Bee Gees.
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Joe Esposito, a close friend and aide of Elvis Presley, has died at the age of 78.