South Africa needed a bonus-point victory and to win by more than 39 points to steal the crown at the very end of the championship.
However, the Springboks failed even to get their bonus point and had to buckle down just to win the game in the face of a Pumas fightback.
It meant that New Zealand, who beat Australia 40-14 in Auckland earlier on Saturday, retained their title and claimed their eighth triumph in 10 tournaments in the Rugby Championship era.
"It's tough but we'll never take a win for granted," Springboks captain Siya Kolisi said. "We knew they were never going to back down. We wish we could have done it (clinch the title), but just grateful to get the win in the end."
Argentina scored three tries in Durban and twice closed the gap to three points – at 17-14 and 24-21 – before the Boks got away in the last 10 minutes.
Australian referee Damon Murphy dished out six yellow cards, four to Argentina, as both sides lost their discipline.
Argentina played half of the first half with 14 men after flankers Marcos Kremer and Juan Martin Gonzalez both went to the bin in the struggle to contain South Africa, and particularly their lineout maul.
That was turned on its head after halftime when Argentina's drives from lineouts caused South Africa all sorts of problems.
Eventually, the Springboks weathered Argentina's ferocious comeback — just as they did last weekend in Buenos Aires — and replacement Kurt-Lee Arendse ran in South Africa's last try after the final hooter.
By then, the All Blacks were the southern hemisphere champions once again.
New Zealand began the tournament with a 26-10 defeat to South Africa and coach Ian Foster on the brink after a startling run of losses for rugby's most successful team.
But the All Blacks pulled it together to continue their dominance of the Rugby Championship since the Tri-Nations expanded to include Argentina in 2012.
South Africa and Australia have one title each since then and Argentina none.
In Durban, the Springboks started with their sights on a grand finale and at 17-0, after tries by No.8 Jasper Wiese and Kolisi, and a penalty from inside his own half by Frans Steyn, it looked on.
But Gonzalo Bertranou's try right on halftime changed the momentum.
Flanker Gonzalez closed the gap even further seven minutes into the second half.
Emiliano Boffelli kicked the conversion from out wide to make it 17-14 and South Africa had to scramble just to end the tournament with a win.
The Springboks had two penalty tries from collapsed mauls near the Argentina line to help. They came either side of Argentina's third try, from Matías Moroni.
Arendse put the seal on it to mark his return from a four-week suspension he picked up in the opening round of the championship.
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