More than 50 others were injured.
Officials say it is unclear what sparked the blaze at the city centre five-storey building, which had been abandoned but was being occupied by homeless people.
South African President Cyril Ramaphosa described the incident as “a great tragedy, felt by families whose loved ones perished in this terrible manner”.
The city of Johannesburg confirmed in a news conference that it owned the building, but said cartels had taken it over.
A spokesman for the emergency services, Robert Mulaudzi, told the BBC that firefighters had been able to bring out some of the occupants.
He said the fire had gutted the building and the search for other victims was continuing.
Speaking separately to Newzroom Afrika, Mr Mulaudzi said that the youngest of the seven children who died was 18 months old.
Disaster management officials are also in the area to help provide relief for surviving residents.
Mr Mulaudzi said the scene would be handed over to the South African police service after emergency services had finished searching for victims.
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Living inside a ‘hijacked building’
“We are moving floor by floor conducting these body recoveries,” Mr Mulaudzi told local broadcaster ENCA.
Mr Ramaphosa said he hopes investigations into the fire will enable communities and authorities to prevent a repeat of such a tragedy.
A video posted by Mr Mulaudzi to the platform X, formerly known as Twitter, showed fire trucks and ambulances outside the building with burnt-out windows.
Photos from the scene showed covered bodies lined up near the burned building.
One woman told journalists she was outside the building searching for her 24-year-old daughter.
“As soon as I heard the building was burning down, I knew I had to run here to come and look for her,” she said.
“Now that I’m here, I’m kept in suspense because I really don’t know what is happening. I don’t get any direction – so I’m actually very anxious, I don’t know if my daughter is alive.”
The building is located in what was formerly a business district in South Africa’s economic hub. It was being used as an informal settlement, Mr Mulaudzi said.
A map locates the scene of the fire in Johannesburg, South Africa
The inner city neighbourhood is infamous for “hijacked” buildings, a term used in South Africa to refer to buildings illegally taken over by people struggling to find housing. Some of those who use the buildings include undocumented migrants, mostly from other African countries.
Mr Malaudzi told the BBC that the building had been abandoned previously, but homeless people had moved in looking for shelter during the current cold winter months.
He added that since it was not a formal accommodation with a lease, the building was not properly looked after, and makeshift structures and debris had made it hard to search for and rescue people.
Lebogang Maile, the politician responsible for housing in the province, said there was a chronic problem with housing in the area, with 1.2 million people looking for somewhere to live.
When asked whether his administration would take responsibility for the tragedy, the Mayor of Johannesburg, Kabelo Gwamanda, said the government was dealing with the issue of cartels hijacking buildings which was taking place across the city.
In a visit to the scene of the deadly fire, Mr Gwamanda said Johannesburg officials would relocate squatters living in similar “hijacked” buildings and turn those buildings into social housing. “We are not going there with brute force,” he told reporters, “we are trying to apply a sensitive strategy.”
Mr Mulaudzi told the BBC that in his 23 years of service, he had “never come across something like this”. The emergency services spokesman added that Johannesburg “must do something about it as a city, working together, to make sure we prevent incidents like this one”.
In the wake of the fire, many South Africans on social media condemned the online xenophobic attacks that some have made against the victims and survivors of the fire.
