The bus crashed through a barrier and caught fire when it hit the ground in the north-eastern Limpopo province.
The passengers were pilgrims travelling from Botswana’s capital Gaborone to an Easter service in the town of Moria.
The vehicle lost control and went off a bridge on the Mmamatlakala mountain pass between Mokopane and Marken, around 300km (190 miles) north of Johannesburg, according to South African public broadcaster SABC.
Rescue operations went on late into Thursday evening, with some of those killed reportedly hard to reach amid the debris.
“Our thoughts and prayers are with you during this difficult time,” she added. “We continue to urge responsible driving at all times with heightened alertness as more people are on our roads this Easter weekend.”
South Africa has a poor road safety record.
In an Easter message released earlier in the day, President Cyril Ramaphosa urged citizens to “do our best to make this a safe Easter”.
It should “not be a time where we sit back and wait to see statistics on tragedy or injuries on our roads”, he added.
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