The voter turnout was estimated at 80 per cent – the highest since South Africa's first multiracial election 15 years ago.
Queues snaked outside polling stations across the country from before dawn until past dusk on Wednesday.
The Independent Electoral Commission (IEC) admitted that there was a shortage of ballots for voters in several areas and many centres had to allow people to vote beyond the 9pm (19:00 GMT) cut-off.
"The response is absolutely overwhelming all over the country," Brigalia Bam, the IEC chairwoman, said.
Al Jazeera's Jane Dutton, reporting earlier from a polling station in Johannesburg, said that officials had resorted to using egg crates after they ran out of ballot boxes.
Many analysts believe the ANC, whose anti-apartheid credentials make it the choice for millions of black voters, will win the elections.
But many voters are also frustrated about corruption, poverty and crime and that, they believe, might cause the party's majority to drop from the nearly 70 per cent it achieved in 2004, to below the two-thirds mark that gives it the right to change the constitution at will.
Al Jazeera's Haru Mutasa, reporting from Cape Town, said many of the country's newspapers on Thursday morning were predicting that the ANC would get their two-thirds majority.
But many people have voiced support for opposition parties after becoming disillusioned with the ANC for taking too long to deliver on promises, she said.
Margaret Nkoane, 57, said in Soweto, a Johannesburg township that symbolised the anti-apartheid struggle, said she "voted for the ANC out of loyalty because my father was active in the struggle".
"But I'm not satisfied with what they've done. People expected jobs but they are still living in shacks."
Still, the ANC is expected to capture enough votes for Zuma, 67, to become president.
If he does win, the son of a housekeeper who spent a decade jailed alongside Nelson Mandela, the country's democracy icon and first black president, would take charge as Africa's biggest economy teeters on the brink of its first recession in 17 years.