Zenani Mandela, 13, died when travelling home after the concert in Soweto, the Nelson Mandela Foundation said. No-one else was hurt.
Zenani was one of Mr Mandela's nine great-grandchildren.
The incident came just hours before the former South African president was due to attend the opening of the World Cup in Johannesburg.
A police spokesman said Zenani had been travelling with Nelson Mandela's former wife, Winnie Madikizela.
"The family has asked for privacy as they mourn this tragedy," the Foundation said in a statement.
Zenani had reportedly celebrated her 13th birthday on 9 June.
Thursday's World Cup concert had drawn tens of thousands to Soweto, and traffic in the area had been busy into the early hours of Friday.
The event featured a cast of international stars - including Colombian singer Shakira, the Black Eyed Peas and Alicia Keys, as well as African stars Amadou & Mariam and Hugh Masekela.
Anti-apartheid icon Mr Mandela, 91, had campaigned for the World Cup to come to South Africa.
Some 350,000 people are expected to visit South Africa for the competition, which is being held in Africa for the first time.