Domingo, 38, is assistant coach to Kirsten, who announced on Friday he would not be renewing his two-year contract which expires at the end of July.
"I know what David Moyes is feeling," said Domingo, comparing the challenge of taking over from a successful coach as being similar to Moyes' appointment to succeed Alex Ferguson as Manchester United manager.
Under Kirsten, South Africa became the number one team in the world Test rankings.
Kirsten's last assignment will be to take the South African team to the ICC Champions Trophy in England and Wales in June. Domingo will be in charge when South Africa play a one-day series in Sri Lanka in July and August.
Nenzani said the terms of Domingo's contract had not yet been negotiated but it was likely to be a two-year appointment, which would include the 2015 Cricket World Cup. He said Domingo would be given a free hand in the appointment of support staff.