Mr Zuma said the South Africans had died in a nine-hour “high-tempo battle” against the “bandits” in Bangui.
South Africa had about 200 troops stationed in the city to block Seleka rebels from seizing power.
Ousted CAR leader Francois Bozize has fled to neighbouring Cameroon, officials there have announced.
President Jacob Zuma says he is “deeply saddened” by events
Following the rebel takeover, the African Union has suspended CAR and imposed sanctions on Seleka’s leaders.
“The council asks all member states to take the measures necessary to completely isolate the authors of this anti-constitutional change of power,” said AU peace and security council chief Ramtane Lamamra, adding that this includes travel restrictions and an asset freeze on Seleka’s leaders.
Rebel leader Michel Djotodia said he would uphold a peace accord that promised elections in three years.
He also said the existing power-sharing government – headed by Prime Minister Nicolas Tiangaye, an opposition figure – would remain in place.
“We are not here to carry out a witch-hunt,” Mr Djotodia told Radio France Internationale (RFI).
Rebel ‘surrender’
Looters and armed gangs roamed the streets of Bangui after Mr Djotodia’s forces captured the presidential palace on Sunday.
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Analysis
Pumza FihlaniBBC News, Johannesburg
Sitting ducks? This is the question on the minds of many South Africans after the deaths of 13 soldiers in fighting with rebels in the troubled Central African Republic.
South Africa has for years prided itself on having one of the most organised armies in Africa but the deaths of these men begs the question: “Could the government have avoided this?”
President Zuma says just over 200 troops were deployed to the CAR in January, half the number that was expected to have been sent to support and train the poorly trained and ill-equipped forces of ousted President Francois Bozize. But officials are now hard-pressed to explain the role the South African soldiers were expected to play in the event of an attack.
Critics say the government may have underestimated the rebel threat and therefore did not react appropriately. Others have questioned why South Africa’s role did not change from trainer to fighter when it became clear weeks ago that the rebels were gaining ground.
There are still many unanswered questions but some have accused the government of poor judgement, saying the incident has not only caused embarrassment for the army, but for South Africa as a key player in peace missions in Africa.
Thousands of people have fled CAR into neighbouring Democratic Republic of Congo and Cameroon, said United Nations High Commission for Refugees (UNHCR) spokesperson Fatoumata Jejeune Kaba.
“Before this weekend, we had already received more than 26,000 refugees who are in the province of the Equator [in DR Congo],” she told BBC Afrique.
Mr Zuma said just over 200 South African troops had battled more than 1,000 rebel fighters.
“They fought a high-tempo battle for nine hours defending the South African military base, until the bandits raised a white flag and asked for a ceasefire,” he said.
“Our soldiers inflicted heavy casualties among the attacking bandit forces.”
Thirteen South African troops were killed, 27 wounded and one was unaccounted for, Mr Zuma said.
‘Mercenaries’
It was the highest death toll South Africa’s army had faced in a single battle since white minority rule ended in 1994, correspondents say.
The BBC’s Milton Nkosi in Johannesburg says Mr Zuma’s government must be worried that one of the most powerful armies in Africa had suffered heavy casualties at the hands of a rebel group.
It was bewildering that South Africa had sent troops to CAR as part of a bilateral agreement with Mr Bozize’s government, rather than through the AU, he says.
The South African National Defence Force Union (Sandu), which represents soldiers, said Mr Zuma should recall troops immediately, “even if it entails launching a military offensive to relieve our troops in distress”, its national secretary Pikkie Greeff said.
But Mr Zuma said South Africa planned to reinforce its troops in CAR, as there was “no reason” to leave.
South Africa sent 200 troops in January out of a planned deployment of 400, Mr Zuma said.
Last week, Seleka denounced the South African forces as “mercenaries” and demanded their withdrawal.
Mr Djotodia told RFI that free and fair elections would be held within three years, as outlined in the peace agreement signed with Mr Bozize in January.
But the deal collapsed last week when Seleka withdrew its members from the government.
The group accused Mr Bozize of running a parallel administration and failing to honour key aspects of the deal, including the release of political prisoners.
UN Secretary-General Ban Ki-moon condemned the rebel seizure of power and called for “the swift restoration of constitutional order”.
As the rebel forces arrived on the outskirts of Bangui on Saturday, France sent an extra 300 troops to back up its 250 soldiers already there to protect an estimated 1,250 French nationals in the country, AFP reports.
CAR, which has a population of about 4.5 million, has been hit by a series of rebellions since independence from France in 1960.