The away side had their kicking game to thank as they were outscored on the night, but had enough experience to get through.
Morne Steyn's penalty reduced the deficit, but the score was 10-3 after 10 minutes with Contepomi kicking one of his own.
All inside a frantic opening 15 minutes South Africa were level with Bjorn Basson's try converted by Steyn.
Argentina timed their next try perfectly, with Marcelo Bosch's converted effort putting them 17-10 up before a penalty from Steyn pulled it to within four at the half.
In the second 40 minutes, the game was decided with three penalties. Steyn's kicking was exceptional and the nine points from his boot gave his side the win.