The Nelson Mandela Foundation said Zenani Mandela died in a one-car accident and no one else was injured.
The incident came just hours before the former South African president was scheduled to attend the opening of the World Cup.
“The family has asked for privacy as they mourn this tragedy,” the foundation said in a statement.
It reportedly added that Mandela will attend the opening ceremony despite the tragedy, but for a much lesser time than he was earlier scheduled to spend.
Mandela, 91, is widely credited with helping South Africa win the World Cup bid in 2004 and South Africans have been hoping he might be able to attend the opening match despite his frail health.
Zenani Mandela, killed two days after her birthday, was one of Mandela’s nine great-grandchildren.
“Zenani Mandela, 13, was travelling in a car returning from the World Cup Kick-Off Concert at Orlando Stadium when the accident occurred,” a statement from the Foundation said.
“There was only one car involved and no one else was injured.”
Mandela’s ex-wife Winnie Madikizela-Mandela was in the car, the South African Press Association quoted police as saying. Police spokesman Noxolo Kweza said the driver had been arrested and police were investigating a case of culpable homicide.
Zenani was a grand-daughter of Zindzi Mandela, Nelson Mandela and Winnie Madikizela-Mandela’s daughter.