Zenani Mandela was one of Nelson Mandela's nine great-grandchildren
The car's driver has been arrested and charged with drink-driving, police say.
The Nelson Mandela Foundation said the tragedy made it inappropriate for the former president, who is 91, to attend the opening ceremony in Johannesburg.
"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," said a statement.
"Madiba [Mr Mandela's clan name] will be there with you in spirit today."
A host of celebrities from the world of politics, business and sport will attend the opening ceremony at Soweto's Soccer City stadium later on Friday, before the competition's opening game between South Africa and Mexico.
Drink-driving charge The car carrying Zenani Mandela overturned as it came off a slip road on the M1 motorway, which travels through Johannesburg city centre, at around 0030 (2230 GMT on Thursday), reports said.
Police say the unnamed male driver, who was reportedly unhurt,
may also face culpable homicide charges.
"He lost control of the vehicle and it collided with a barricade," police spokeswoman Edna Mamonyane told the Associated Press. "The Metro police found that he was drunk."
The driver was expected to appear in court for a preliminary hearing later on Friday.
Early reports quoted police as saying Nelson Mandela's former wife, Winnie Madikizela, had been in the car when the accident happened.
The Foundation told the BBC Ms Madikizela-Mandela was not in the car - which was the only vehicle involved in the crash - but that she had been taken to hospital because of her shock at the news of Zenani's death.
Foundation spokesman Sello Hating said she had been released from hospital after a couple of hours and returned home.
"The family has asked for privacy as they mourn this tragedy," said the Foundation in a statement.
In a letter of condolence to Mr Mandela, Fifa President Sepp Blatter described the young girl's death as "unspeakably tragic".
Zenani had celebrated her 13th birthday on 9 June.
South Africa has one of the world's worst road safety records, and there are nearly 50 deaths each day on roads across the country.
Heavy traffic Anti-apartheid icon Mr Mandela 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.
More than 1,500 performers are set to take part in Friday's opening ceremony, which will be broadcast in 200 countries around the globe.
All the major roads leading to the Soccer City stadium were under severe pressure several hours before the ceremony was due to start, said the BBC's Pumza Fihlani, as fans in thousands of cars tried to make their way into Soweto.
Roads to nearby carparks for fans were gridlocked, our correspondent added.
Thursday's World Cup concert had drawn tens of thousands to Soweto's Orlando Stadium, 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.
Nelson Mandela is to miss
the World Cup opening ceremony after the death of his
great-granddaughter, Zenani, in a car crash, his office says.
The 13-year-old died when the car taking her home from a Soweto
concert on the eve of the World Cup overturned.
The car's driver has been arrested and charged with drink-driving, police say.
The Nelson Mandela Foundation said the tragedy made it inappropriate for the former president, who is 91, to attend the opening ceremony in Johannesburg.
"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," said a statement.
"Madiba [Mr Mandela's clan name] will be there with you in spirit today."
A host of celebrities from the world of politics, business and sport will attend the opening ceremony at Soweto's Soccer City stadium later on Friday, before the competition's opening game between South Africa and Mexico.
Drink-driving charge The car carrying Zenani Mandela overturned as it came off a slip road on the M1 motorway, which travels through Johannesburg city centre, at around 0030 (2230 GMT on Thursday), reports said.
Nelson Mandela Foundation spokesman Sello
Hatang: "Mandela will be there with you in spirit today"
"He lost control of the vehicle and it collided with a barricade," police spokeswoman Edna Mamonyane told the Associated Press. "The Metro police found that he was drunk."
The driver was expected to appear in court for a preliminary hearing later on Friday.
Early reports quoted police as saying Nelson Mandela's former wife, Winnie Madikizela, had been in the car when the accident happened.
The Foundation told the BBC Ms Madikizela-Mandela was not in the car - which was the only vehicle involved in the crash - but that she had been taken to hospital because of her shock at the news of Zenani's death.
Foundation spokesman Sello Hating said she had been released from hospital after a couple of hours and returned home.
"The family has asked for privacy as they mourn this tragedy," said the Foundation in a statement.
In a letter of condolence to Mr Mandela, Fifa President Sepp Blatter described the young girl's death as "unspeakably tragic".
Zenani had celebrated her 13th birthday on 9 June.
South Africa has one of the world's worst road safety records, and there are nearly 50 deaths each day on roads across the country.
Heavy traffic Anti-apartheid icon Mr Mandela had campaigned for the World Cup to come to South Africa.
Dangerous roads
Continue reading the main story Estimated road traffic death-rate per 100,000 people- Eritrea: 48
- Egypt: 41.6
- Iraq: 38
- South Africa: 33.2
- Nigeria: 32.3
- Brazil: 18.3
- India: 16.8
- US: 13.9
- UK: 5.4
- Japan: 5.0
More than 1,500 performers are set to take part in Friday's opening ceremony, which will be broadcast in 200 countries around the globe.
All the major roads leading to the Soccer City stadium were under severe pressure several hours before the ceremony was due to start, said the BBC's Pumza Fihlani, as fans in thousands of cars tried to make their way into Soweto.
Roads to nearby carparks for fans were gridlocked, our correspondent added.
Thursday's World Cup concert had drawn tens of thousands to Soweto's Orlando Stadium, 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.