She will also visit Nigeria and Kenya during the three-day trade mission.
The visit - which will see PM May meet the presidents of all three countries - aims to deepen economic and trade ties with growing African economies ahead of Britain leaving the European Union in 2019.
May, who was accompanied by a delegation of British business executives, also said Britain would work with African states to tackle insecurity and migration by creating jobs.
May, who assumed office two years ago, met with pupils at a Cape Town school, breaking into a spontaneous dance as they greeted her to traditional South African music.
The PM said it was her view that foreign aid "works", adding: "We will remain a global champion for aid spending, humanitarian relief and worldwide development".
Britain's overseas aid budget totalled £13.9 billion in 2017, an increase of £555 million in 2016 and in line with the legal commitment to invest 0.7 per cent of national income in this area.
Data from the UN Conference on Trade and Development shows United Kingdom direct investment in Africa stood at $55bn (£42.7bn) in 2016, compared to $57bn (£44.3bn) from the US.
Former foreign minister Boris Johnson, whose July departure from the cabinet brought May's government to the brink, said in his resignation speech that May's current Brexit policy would hamper London's ability to strike independent trade deals.
Mrs May's visit to Nairobi will mark the first by a British prime minister to Kenya since Margaret Thatcher in 1988.
Mrs. May went on to say she wants a healthy business relationship with Africa.
At the press conference, May also voiced support for South Africa's controversial land reform plans.
In Nigeria, the UK Prime Minister will meet President Muhammadu Buhari in Abuja and spend time in Lagos meeting victims of modern slavery.
The last time a British prime minister visited sub-Saharan Africa was in December 2013 when David Cameron attended Nelson Mandela's funeral.
On Thursday, she will meet Kenyan President Uhuru Kenyatta, shortly after his return from seeing US President Donald Trump in Washington and before he travels to China to meet President Xi Jinping.