Nene made the apology after detailing his meetings with the Guptas at the state capture commission of inquiry last week - until then‚ the meetings had remained secret.
Mboweni, who went into exile during apartheid, trained as an economist and served for four years as labour minister in former President Nelson Mandela's cabinet after apartheid ended in 1994.
South African Finance Minister Nhlanhla Nene resigned Tuesday over undisclosed meetings with a business family at the heart of a corruption scandal, dealing a blow to President Cyril Ramaphosa.
Nene faced calls to resign after he admitted to visiting the Gupta brothers, friends of the former president Jacob Zuma who have been accused of high-level influence-peddling, and failing to disclose the meetings earlier.
"I have made a decision to accept his resignation", Ramaphosa told a televised press briefing in Cape Town.
While Mboweni has a good reputation, doubts will continue to linger over Ramaphosa's ability to clean up after Zuma's scandal-marred tenure, according to Guillaume Tresca, a senior emerging markets strategist at Credit Agricole CIB from Montrouge, France.
Nine years after leaving his post as governor of the South African Reserve Bank, Tito Mboweni is has been appointed as the nation's finance minister and faces the unenviable task of reviving an economy that's mired in recession.
This was Nene's second stint as finance minister. "I deeply regret these lapses and beg your forgiveness".
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The rand gained 0.9% to R14.72/$ by 5.25pm in Johannesburg, reversing an earlier decline of as much as 1.4%.
The ANC forced Zuma to quit as president in February, clearing the way for his 65-year-old deputy and party leader Ramaphosa to take over.
He must also reassure investors and credit-rating companies of credible plans to stabilise the debt and revive growth in the mid-term budget on October 24.
At the graft inquiry, Nene accused Zuma of pushing policies created to benefit the Guptas, including a massive nuclear power expansion programme.
Commenting on the state capture inquiry, Ramaphosa said it was critical that the commission had the means and opportunity to effectively fulfil its mandate.
The main opposition Democratic Alliance party welcomed Mboweni's appointment, saying he was a safe pair of hands.
The rand firmed, moments before Mboweni, was announced Finance Minister at a ceremony in Cape Town.
"The Medium-Term Budget Policy Statement‚ which is supposed to be a statement to build confidence among all important economic role players‚ cannot‚ and should not‚ be delivered by a minister who was part of the Gupta criminal syndicate".