Both sides scored two tries as the Pumas, who led 13-3 at halftime, beat the Springboks for the second time in 24 meetings and the second in as many seasons.
It was Argentina's third win in the championship since joining the elite southern hemisphere tournament in 2015, having also beaten Australia once.
The Springboks, for whom winger Bryan Habana grabbed a record 65th try on Saturday, came from behind to beat the Pumas in Nelspruit last weekend and were within minutes of doing it again.