Los Pumas were given a glimmer of hope when Tomas Cubelli- yellow-carded earlier in the game - slotted over a drop goal, but the Springboks remained resilient until the dying seconds when Juan Pablo Orlandi touched down.
Relive Sports Mole's live text coverage below.
SOUTH AFRICA XV: Willie le Roux; JP Pietersen, Jesse Kriel, Damian de Allende, Bryan Habana; Handre Pollard, Ruan Pienaar; Tendai Mtawarira, Bismarck du Plessis, Frans Malherbe, Eben Etzebeth, Victor Matfield (capt), Francois Louw, Schalk Burger, Duane Vermeulen
ARGENTINA XV: Lucas Gonzalez Amorosino; Santiago Cordero, Matias Moroni, Jeronimo de la Fuente, Horacio Agulla; Nicolas Sanchez (capt), Tomas Cubelli; Juan Figallo, Julian Montoya, Ramiro Herrera; Matias Alemanno, Tomas Lavanini; Javier Ortega Desio, Juan Fernandez Lobbe, Juan Manuel Leguizamon
© Getty Images
"It's the game no-one wants to be involved in, but these two sides have such pride in the jersey that I'm still expecting a full-blooded, entertaining contest tonight. Argentina have drawn so many plaudits for the way that they have performed so far at this World Cup, while South Africa have looked fallible at times.
"Another third-placed finish for certainly be a good achievement for the Pumas, and they can take great heart from beating the Springboks already this year - the first time that they have ever managed to do so. Even so, their injuries could prove too much to overcome tonight - they are missing their captain, playmaker and best try-scorer, while South Africa have named an almost full-strength side."
"I would have loved to have seen these two go head to head both with full-strength sides. This Argentina team have evolved their style ever since being included in the Rugby Championship, and they are becoming more and more of an attacking threat as a result. Indeed, only the All Blacks have scored more points and more tries than the Pumas at this World Cup. However, they have been ravaged by injuries ahead of this match and that is likely to have a fatal impact on their chances. I'm going for a South Africa win by nine."
© Getty Images
A fifth score of the World Cup for JP Pietersen, who nicely finished off the Springboks' first real attack. A nice gap was found out wide after initially failing to go down under the posts, and Pietersen did well in a one-on-one situation to stretch over. No mistake from the follow-up by Handre Pollard.
"Habana was inches away from breaking the record there, but despite that one not being awarded it has still been a good start from the Springboks and a bit of a nightmare for Argentina. A yellow card and converted try inside the opening minutes is not what they would have wanted here."
The Springboks looking particularly dangerous now, without being able to add a second try to their tally. They do have three more points via the boot of Pollard, though, coming after an obstruction 25 metres from the whitewash.
Pollard makes no mistake from the pen, so that really was a poor error of judgement from Sanchez. Unnecessary does not come close.
That second try is still alluding Meyer's men, but their tally is ticking along nicely as Pollard boots over three more points to become the tournament's top scorer. More slow play at blame for Los Pumas there.
"So far so good for the Springboks, then. Argentina's attack has shown flashes, but this isn't the first time that South Africa's defence has shown resilience at this World Cup. They lead by 16 at the break and it would be some achievement if Argentina come back from this - a record-breaking one, in fact. All eyes are on Bryan Habana, who has had four or five nearly moments already as he goes in search of that unprecedented 16th Rugby World Cup try."
Well that has certainly lifted the atmosphere! Sanchez, from out of nowhere, slots the ball nicely through to sticks with a drop goal from 40 metres out. Right down the middle, too.
Breathing space for South Africa after that early setback in this second half, as Etzebeth responds superbly to touch down in the corner. Kriel and Habana were involved in the build-up, before Etzebeth was given the honour of making it 21-3. Pollard was wayward from the follow-up attempt.
Too many offsides from Argentina, which forces the referee to have a word. Pollard profits from it, slotting over another pen to make that advantage look even stronger at this stage.
The penalty advantage comes back for Argentina, giving them a chance to reflect. Still no way through the South Africa defence, so Sanchez this time kicks over to reduce the lead somewhat.
© Getty Images
"Argentina keep on battering away, but South Africa deserve great credit for keeping them at bay. They have missed a fair few tackles, but all in all they have once again defended admirably. There were questions whether they would be up for this one as much as the Argentinians, but they are rubbishing such speculation with a professional display here."
South Africa opt to take the pen on, despite holding this hefty advantage, and those not backing the Springboks inside the Olympic Stadium are delighted to see Pollard strike the post.
How about that. They have been pushing and pushing and pushing, and finally they have something to show for their efforts. Orlandi stretched over from a metre or so out to provide even more joy to a proud nation.
"The biggest cheer of the night greets that late try as Argentina finally get on the board. Despite the result, it was the Argentine fans making all the noise towards the end of that match. It was something of an anticlimax on the field as far as the South Americans are concerned, but the odds were always stacked against them today and they have had a fine World Cup on the whole.
"South Africa, meanwhile, would have been fearing a lot worse than third place after their defeat to Japan in the opening game so, while failing to make the final will be a lingering disappointment, they can be relatively happy with how they have responded to that stunning loss. There are big questions remaining over how they will cope with the loss of messrs Matfield, Burger, De Villiers, Du Preez and maybe even Habana, if not tonight then in the next couple of years, while the future of Argentina looks very bright indeed."