Some 20,000 polling stations were being used for more than 23 million registered voters.
Polls were due to close at 2100 local time (1900 GMT), but voting was extended in some areas where people had queued for many hours.
The BBC's Andrew Walker at the electoral commission's headquarters in Pretoria reported that officials there said if voters were in a queue at 2100, then they would be able to vote, no matter how long it took.
The governing ANC and the official opposition, the Democratic Alliance, had both asked for an extension of voting because of polling problems.
Counting will begin as soon as polling stations around the country close.
"The response is absolutely overwhelming all over the country," said Independent Electoral Commission (IEC) chairwoman Brigalia Bam earlier in the day.
But by late afternoon, reports emerged that voting stations in Gauteng, the Free State and Mpumalanga were running out of ballot papers.
Hear what some voters waiting to cast their ballots think
After casting his vote in Johannesburg, Mr Mbeki did not confirm the speculation, saying: "People should vote for who they want, not out of fear, but for the party they believe will deliver the South Africa they want."
Voting in his home state of KwaZulu-Natal, Mr Zuma told cheering supporters: "When I grew up, I did not know that this day would come.
"This makes me feel great and it's a feeling far different from the one that we had under the apartheid government."
Archbishop Desmond Tutu, who has described Mr Zuma as an unfit president, cast his vote in Cape Town.
"Because of the role the ANC has played in the struggle, and in the first years of our freedom, most people would have tended to vote ANC," he said.
"Now, it is no longer quite so straightforward. People are asking questions, which is a good thing, I mean, that is what a democracy is."
South Africa's last white president, FW de Klerk, cast his vote in Cape Town and told reporters he believed the country's democracy would be healthier because of the split in the ANC.
"I think our (party political) situation becomes more competitive," he said.
Charges of corruption against Mr Zuma were dropped just two weeks before the poll, after state prosecutors said there had been political interference in the case.
'Voting for change'
Many of the new voters are young people who have little memory of the struggle to end white minority rule, which brought the ANC to power.
One first-time voter, Bhekisa, was among the first at the polling station in Johannesburg City Hall.
"It's so cold today, you can see it's freezing," he said. "But I am excited because I am here."
Zille responds to charges of running a negative campaign:
Analysts say Cope's emergence energised the early stages of the election campaign, but the party's popularity seems to have diminished in recent weeks.
Cope fielded a relatively unknown presidential candidate, former Bishop Mvume Dandala, who analysts say has struggled to make an impact.
Some say the real battle is between Cope and the Democratic Alliance - for second place.
Neither party has ruled out entering into a coalition after the election.
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