Zenani Mandela-Dlamini is the oldest daughter from the former president’s marriage to Winnie Madikizela-Mandela.
Argentina said her appointment represented the importance South Africa attached to their relationship.
Ms Mandela-Dlamini, 53, was a child when her father was imprisoned for 27 years by the white minority government.
She was not allowed to visit him in jail until she was 16 years old, the Associated Press news agency reports.
A graduate of the University of Boston, Ms Mandela-Dlamini was once married to a Swazi prince.
Mr Mandela became South Africa’s first black and democratically elected president in 1994, stepping down after one term in office.
The Nobel Peace Prize laureate, who turns 94 this month, retired from public life eight years ago – his last public appearance was at the World Cup in South Africa in 2010.
“During the presidency of Nelson Mandela, recognised as a global figure for his fight against apartheid and for human rights, Zenani Dlamini accompanied her father to act as first lady of her country,” Reuters news agency quotes a statement from Argentina’s foreign ministry as saying.
According to the AFP news agency, bilateral trade between South Africa and Argentina is worth about $1.3bn (£830m).