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 chargeThe 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”
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.”
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.
Zenani had celebrated her 13th birthday on 9 June.
Heavy trafficSouth Africa has one of the world’s worst road safety records, and there are nearly 50 deaths each day on roads across the country.
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.
More than 1,500 performers are set to take part in Friday’s opening ceremony, which will be broadcast in some 200 countries around the globe.