The All Blacks' dominant 40-14 defeat of Australia had set the Springboks the sizeable task of securing a bonus-point win by at least 39 points to be crowned champions in Durban.
Ian Foster's side could ultimately watch on with some degree of comfort early on Sunday morning back home, as the world champion could not prevent it from claiming a fifth title in six championships and had to settle for second place.
An 11th-minute Springboks try was ruled out as Eben Etzebeth had stripped the ball from team-mate Siya Kolisi in an offside position, although their pressure soon told with a yellow card for Marcos Kremer.
South Africa made the most of its numerical advantage when Jasper Wiese scored the first and surely easiest try of his international career, walking the ball over at the back of a scrum.
No sooner had Kremer returned than Juan Martin Gonzalez headed for the sin bin, and it was captain Kolisi's turn to capitalise as he powered through for the second, before Frans Steyn dispatched a long-range penalty.
But Gonzalo Bertranou lunged for the line on the stroke of half-time to silence the Kings Park crown, then Gonzalez raced down the left for the first try of the second period.
Argentina had designs on winning the match, yet conceded pivotal penalty tries either side of Matias Moroni's score in a half that saw two yellow cards for each side, with the Springboks forced to scrap for their victory as Etzebeth and Faf de Klerk were off the field at the same time.
Kurt-Lee Arendse's last-gasp try at least allowed South Africa to finish with a flourish, with the Pumas consigned to finishing bottom of the table.