The Gupta family, who are Indian expatriates, are accused of working with former President Jacob Zuma.
Ramaphosa accepted his resignation and thanked Nene for his time in office.
Both Zuma and the Guptas - who are at the heart of the inquiry into so-called "state capture" - have denied any wrongdoing.
"It is a measure of his character and his commitment to the national interest that he has taken this decision to resign in the wake of errors of judgment‚ even though he has not been implicated in acts of wrongdoing".
Tito Mboweni, a former Reserve Bank governor, will replace him. He made a public apology about the matter on Friday.
The Gupta brothers - Ajay, Atul and Rajesh - have been accused of using their friendship with Zuma, Ramaphosa's predecessor, to syphon off billions of rand in state funds and of inappropriately influencing cabinet appointments.
Ramaphosa has vowed to crack down on corruption as he tries to revive the economy and boost declining support for the ruling ANC party ahead of elections next year.
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.
Mr Nene met the Guptas at their businesses and home at least six times between 2009 and 2014.
"The initial selling reaction on the rand on reports that its finance minister has asked to be sacked will encourage investor caution that South Africa could be in store for another round of political risk", said Jameel Ahmad, head of currency strategy at FXTM.
Asked repeatedly at a media event in Cape Town whether Nene still enjoyed his confidence and what the future might hold for the finance minister, President Ramaphosa made light of the questions and pretended there was something wrong with his hearing. His biggest achievement at the time was building the nation's foreign-exchange reserves to nearly $40bn from less than $10bn.
After leaving the Reserve Bank, Mboweni went into business, acting as an adviser in South Africa for Goldman Sachs Group and served as chairman of AngloGold Ashanti and on the boards of other companies.
By accepting Mr. Nene's resignation, Mr. Ramaphosa is setting a tough new standard for corruption issues.