De Bruin stepped down from his role for "personal and medical reasons" after a win over Argentina last weekend, which sealed a first Rugby Championship triumph for a decade.
South Africa director of rugby Rassie Erasmus said: "Swys made a great contribution to the squad in his time with us and I'd like to thank him on behalf of his fellow management and the players he has worked with in the past 12 months.
"He brought a fresh perspective to one facet of the game and he'll be hard to replace.
"I'd also like to thank the Lions for releasing him to us and SA Rugby for supporting his appointment."
Erasmus will weigh up his options before appointing a short-term replacement for De Bruin.
"This was not something we were expecting but our planning has been thorough and it's something we can manage," he added.
South Africa faces the Pumas again at Loftus Versfeld on Sunday (AEST), five weeks before taking on defending champion New Zealand in a blockbuster World Cup clash.