The Pumas raced into an early lead before being pegged back by the Boks before the break, with Steyn's composed performance sealing the win in the closing stages.
Argentina started on the front foot and they took an early lead when flanker Juan Manuel Leguizamon barged over in the second minute after a lineout close to the Boks line.
Felipe Contepomi added the extras from the touchline to give the hosts a 7-0 lead, but Springboks stand-off Steyn soon cut the deficit when he slotted a penalty after the Pumas were penalised at the scrum.
Contepomi restored the lead with a 35-metre penalty in the 11th minute but there was soon action at the other end, with Bjorn Basson going over in the corner for South Africa following some great hands from the backs on the basis of a solid scrum.
Steyn converted to level the scores at 10-10 after 15 minutes.
The game then settled down but Argentina claimed a lead at half-time when centre Marcelo Bosch barged past JJ Engelbrecht and Jannie du Plessis in midfield to claim his second Test try.
Contepomi added a simple conversion, before another Steyn penalty on the stroke of half-time saw the Pumas lead 17-13.
The second half started with an element of controversy as Argentina were penalised and the TMO checked for a possible eye gouge from flanker Pablo Matera on Francois Louw, but there was no clear evidence, with Steyn staying focused to kick another three points.
The Pumas were then dealt a blow as Contepomi had to be replaced by Santiago Fernandez after he looked to be in discomfort following a knock, before tempers then started to flare, with Springboks lock Eben Etzebeth claiming he was bitten.
Again, there was no clear television evidence, and Bosch lined up a penalty attempt for Argentina, which went wide.
With only one point in it, the pressure was beginning to tell, but when Argentina collapsed a maul, Steyn made no mistake as he kicked the resulting penalty as the Springboks claimed a two-point advantage.
South Africa were then awarded another penalty, and again Steyn did not miss as his side ran out 22-17 winners.
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- Felipe Contepomi