"This party is an elephant," Zuma said of the ANC. "You cannot actually topple an elephant," he told a sea of cheering supporters clad in the party colours of yellow, green and black.
The Congress of the People (Cope) party - formed by ANC dissidents with the aim of posing a challenge to the ANC's post-apartheid dominance - won eight per cent of votes counted in the latest tally.
The main opposition party, the Democratic Alliance (DA), was on 16 per cent, with full results due to be announced later on Friday.
The DA pulled ahead of the ANC in the Western Cape province - South Africa's main tourist destination - which is currently controlled by the ANC.
Helen Zille, the leader of the DA, said: "I feel very good about the national results ... We are just above 50 per cent in the Western Cape. That is what we were hoping for ... it means we have doubled our numbers since last time."
Speaking to Al Jazeera, she said: "Cabinet has just passed and approved legislation to change the constitution if the ANC gets two-thirds.
"They have approved the bill that will change the constitution to reduce the powers of local governments very severely."
Adam Habib, the vice chancellor of the University of Johannesburg, told Al Jazeera: "The ANC has had a two-thirds majority for at least six or seven years now, and they have not amended the constitution.
"There have been 18 administrative changes, but nothing that violates the spirit of the constitution, and all of those changes were supported by the opposition parties themselves."
Ebrahim Fakir, an analyst from the Electoral Institute of Southern Africa, said: "The debate has always been whether the ANC was going to get two-thirds or fall below the two-thirds threshold. They were always going to win.
"At this stage, the race for second and third, which is what really I think everyone was really watching over the past two days, is far from settled."
A record 80 per cent of South Africans turned out at nearly 20,000 polling stations on Tuesday, causing ballot shortages and overflowing boxes.
The long lines of voters recalled images of South Africa's first democratic elections in 1994, when Nelson Mandela became president following the end of apartheid.
Despite the public enthusiasm for the polls, The Star
newspaper cautioned that South Africa faced immense challenges made more difficult in the global economic crisis.
"Millions of our citizens remain mired in poverty, our health services are inequitable and our education system is a national tragedy," it said.
"We have heard the campaign talk - now we want a new patriotism and action."
The poll was conducted largely peacefully, but a Cope party official in the Eastern Cape was shot dead by three armed men in an attack on his home.
Zuma has campaigned on a pro-poor ticket with promises of improved public services, but will enter office as South Africa slides towards recession.
Despite the gains since apartheid, public frustrations are growing, with unemployment estimated at 40 per cent.
Crime is soaring and South Africa has the world's largest caseload of people with HIV/Aids.