Graham Briggs, the chief executive officer of Harmony Gold, told Reuters news agency that he was confident all the miners would be lifted to the surface in a small elevator.
"There have been no injuries or deaths ... It may take as long as 10 hours. The cage [lift] is travelling fairly slowly to avoid risks ... I'm very confident all will come out," he said.
Buyelwa Sonjica, South Africa's minerals and energy minister, arrived at the mine on Thursday morning and inspected the scene before meeting with mine officials amid complaints by unions over safety standards.
"This particular mine will be closed for between three and six weeks until the [lift] is repaired. We will prosecute if there is proof of negligence," Sonjica said.
Patrice Motsepe, chairman of Harmony Gold, said the safety of the miners was the first priority.
"Obviously we want everyone out as quickly as humanly possible but if we speed it up more than we are, we run the risk of compromising the safety of our employees," he said.
Nhlauhla Mtshali, a doctor treating those brought to the surface, said that two people had been taken to hospital although no one was in a serious condition.
"Three have collapsed from dehydration, two are receiving treatment in the hospital," she said.
"Five people have complained of cramps and there are two paramedics down on the ground helping people into the lift."
NUM said it suspected negligence and Harmony's practice of mining 24 hours per day caused the accident.
"We suspect negligence. Because of continuous operations there is no time to make adequate checks," Senzeni Zokwana, NUM's president, said.
Briggs said production at the mine had been halted and would remain shut down until an investigation had been carried out and the damage repaired.
The miners were caught after an air pipe broke off and hurtled down the shaft propelled by compressed air, damaging steelwork and severing an electrical cable carrying power to the main lift, he said.
Rescuers were in contact with the trapped miners and clean air and water were being pumped down to them, he said.
Harmony bought the Elandsrand mine and nearby Deelkraal operations from rival AngloGold Ashanti in 2001.
At the time production was declining and Harmony saw potential in digging a new mine underneath the old one.
Harmony, which employs around 44,000 people and produced 2.4 million ounces of gold in 2006, expects to complete the new Elandsrand mine by 2011 and to mine it for a further 18 years.
'In good health'
The first miners at Elandsrand emerged at around 2330 GMT on Thursday, after being trapped 2.2km underground for at least 15 hours.
Although those rescued so far were all in good health, Lesiba Seshoka, a spokesman for the NUM, told Al Jazeera he was worried for those still trapped.
"We are very worried because ... they might be suffocating," he said.
May 2001: Explosion at Beatrix gold mine near Johannesburg kills 12.
July 1999: Methane gas explosion at Mponeng gold mine at Carleton, southwest of Johannesburg, kills 19.
May 1995: Elevator falls down shaft at Vall Reef gold mine, southwest of Johannesburg, kills 105.
April 1987: Methane gas explosion at coal in Ermelo, east of Johannesburg, kills 30.
September 1986: Fire at Kinross gold mine, near Johannesburg, kills 177.
September 1983: Methane gas explosion at Hlobane coal mine in eastern Kwa-Zulu Natal Province kills 64.
January, 1960: Shaft collapse at coal mine in Coalbrook in eastern Kwa-Zulu Natal province kills 437.
Seshoka said the miners were trapped in a cramped space where temperatures could reach 30 to 40 degrees Celsius.
Peter Bailey, another NUM official, told Al Jazeera: "The unfortunate situation is that nobody really recognises the emotional trauma that people are going through under these circumstances, because many of these mine workers cannot get over a situation like this."
Al Jazeera also spoke to Amelia Soares, a Harmony Gold spokerperson, who said: "The mine happens to be one of our new projects, so it is well ventilated. There are water points at all the levels and also medical facilities, for the workers".
She said: "Once we get the rock-and-ventilation shaft going, then we will be able to hoist 300 people per hour.
"We have also sent down paramedics to ensure that everybody is in good health and that no medication is required. Management will ensure that oxygen is pumped throughout the night until everyone is on the surface.
Gold mine operations in South Africa have come under scrutiny over the past few months following a series of accidents as gold producers mine ever deeper to offset lower production and reap the benefits of a sharply higher bullion price.
Gold output in South Africa, the world's biggest gold producer, has tumbled by over 50 per cent over the past decade.
High-grade mines are running out of ore and companies are grappling with more difficult and high-cost underground operations.
South African gold mines are the deepest in the world and unions have often criticised companies for not doing enough to ensure workers' safety.
The government briefly closed an AngloGold Ashanti mine in July after two miners were killed in a rock fall.