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The crash happened late on the evening of Feb. 1 during heavy rain but was only linked to Duduzane this weekend, just one day after his father announced a general election on May 7.
If Duduzane is found to have been driving recklessly, the incident will be another embarrassment to Zuma, who is already fighting a scandal over a 206 million rand ($18.6 million) state-funded security upgrade to his private home.
Police confirmed the details of the accident but declined to name the driver or victims, in line with common practice in South Africa.
"One person died and two are still in hospital. We are investigating culpable homicide," Johannesburg police spokesman Neville Malila said. The investigation might take two to three weeks to complete, he added.
The Sunday Times newspaper named the dead person as 30-year-old Zimbabwean Phumzile Dube. It quoted her husband, Themba, criticising Duduzane for not contacting the family or offering to help pay for her funeral.
"Zuma killed my wife and I want justice," Dube told the newspaper. "It doesn't matter whether he is the president's son - justice doesn't have eyes. Everybody is equal before the law."
The news report said Duduzane, a wealthy businessman, had not been breathalyzed.
The opposition Democratic Alliance (DA) urged Zuma and his government - which is likely to win re-election in May - to assure South Africans that there would be no interference in the investigation.
"The DA will closely monitor this case to ensure that there is no cover-up," shadow police minister Dianne Kohler Barnard said in a statement. "No one, including the Zumas, is above the law."
A government spokesmen declined to comment and Reuters could not reach Duduzane. The Sunday Times and Sunday World newspapers said his mobile phone rang unanswered.