"The ANC invites fellow South Africans to attend this celebration to share the joy of having achieved a strong mandate to consolidate the gains made during the past 15 years," the party said in a statement.
Several thousand ANC supporters were gathering there ahead of an expected address by Mr Zuma, some singing and dancing.
Monday's poll was the country's fourth, and most competitive, general election since the end of apartheid 15 years ago.
Turnout was comparable to the last general election in 2004, when it was 76% and the ANC won 70% of the vote.
Voting was extended in some areas because of the long queues and a shortage of some polling materials.
Results are still expected from many big townships, where the ANC is expected to do well, but the party appears to have suffered a reverse in the area around Cape Town.
With many votes still to be counted, it was already clear that about half of the Western Cape province's voters have opted for the DA, led by Helen Zille.
Karen Allen reports from the Election Commission in Pretoria
This is a considerable setback for Mr Zuma whose party currently controls all the country's nine provinces and could justifiably claim to speak for all parts of South Africa, BBC Africa analyst Martin Plaut reports.
It is now clear that the DA will take this province even if it has to do so in coalition with another party, he adds.
Ms Zille is already mayor of Cape Town and draws much of her support from the area's large white and mixed-race communities.
Before the election, she urged South Africans to vote to prevent the ANC retaining its two-thirds majority, saying the party might then change the constitution to prevent new charges being brought against Mr Zuma.
Mr Zuma has always denied having any such plans.
Second place fight
In Eastern Cape province, where Cope was expected to do well, the new party was being given just 13% of the vote.
Served 10 years for ANC membership during apartheid era
Regarded as a populist who appeals to the left
Dogged by corruption and sex scandals
Aged 67; Zulu; known popularly as JZ; has at least two wives
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