Miners were trapped underground on Wednesday (local time) after an electric feeder cable connected to the mine lift was severed.
The first miners emerged this morning (Australian time) after being trapped 2.2km underground for more than 15 hours.
Rescuers have used an adjacent lift to bring the workers to the surface.
Harmony Gold chief executive Graham Briggs told Reuters there had been no injuries or deaths, however the process of getting all miners to the surface may take up to 10 hours as the lift was travelling slowly to avoid risks.
Briggs added production at Elandsrand would remain offline until the damage had been repaired and an investigation had been carried out.
Harmony, the world's fifth biggest gold producer, has operations in South Africa, Australia and Papua New Guinea.
Its Australian operations include the South Kal tenements, which lie between Kalgoorlie and Kambalda, and Mt Magnet operations in Western Australia.