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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The incident involving a black Ford Ka happened on St Fillans Road at about 21:00 on Saturday.
The girl is being treated at Ninewells Hospital in Dundee.
Police have asked for anyone who may have witnessed the collision, or who saw a black Ford Ka in the area at the time, to come forward.

Police said Alexander Protopopov, acting deputy chief of Russia's prison service, oversaw the dismantling of the road in the far-northern Komi region.
He then sold off its 7,000 reinforced concrete slabs for personal profit, they added.
Officials believe the scheme cost the government more than 6m roubles ($79,000; Â£54,000).
The road was "dismantled and driven away" over the period of more than a year, between 2014 and 2015, the Investigative Committee said in a statement quoted by the AFP news agency.
The concrete slabs were then used by a commercial company which also sold them on for a profit, it added.
Mr Protopopov was the head of Komi region's prison service from 2010 to 2015, and won awards such as a medal for creating "spiritual unity", AFP reports.
He was arrested on Wednesday and faces charges of misappropriating of state property while using his official position, which could lead to 10 years in jail, the agency adds.
Another prison service official, the head of a penal colony and a businessman are also under investigation.
Komi Republic is a huge region in Russia's far north, much of it forested. It has significant natural resources, including oil, gas and timber.
Road construction is one of the most corrupt sectors in Russia, with costs much higher than on comparable projects in other countries, AFP reports.
The construction of a mountain highway for the Sochi Winter Olympic Games cost about $8bn (Â£5.6bn), with Russian media claiming it would cost the same to slather the 48km road with black caviar.
Joe Esposito, a close friend and aide of Elvis Presley, has died at the age of 78.
[NEXT_CONCEPT]
An eight-year-old girl is in a critical condition after being hit by a car in Dundee.
[NEXT_CONCEPT]
A senior prison official has been detained in Russia accused of stealing a 50km (31-mile) length of highway.