The driver of the car was drunk and police have arrested him and are investigating charges of culpable homicide, a police spokeswoman said.
Mandela to be absent
Mandela, 91, helped the country win the 2010 World Cup bid in 2004 and South Africans have been hoping he might be able to attend the opening match despite his frail health.
But he will not be attending the opening celebrations or the opening match in Johannesburg because of the family tragedy.
Sello Hatang, a spokesman for the Mandela Foundation, said that Mandela would not be attending the celebrations.
"We are sure that South Africans and people all over the world will stand in solidarity with Mr Mandela and his family in the aftermath of this tragedy," Hatang said.
Mandela, who turns 92 on July 18, has largely retired from public life but was inundated with requests for meetings as visitors flocked to the country for the football tournament.
"Mandela's diary will remain closed for the duration of the World Cup," his spokesman said, adding that the former president "remains in good health".
Mandela, a Nobel peace prize winner, spent 27 years in prison during the fight against Apartheid.
He has called the month-long World Cup "one of the greatest" events in Africa's history.
"The tournament symbolises the power of football to bring people together from all over the world, regardless of language, the colour of one's skin, political or religious persuasion," he said.