The world champion Springboks, who had the chance to win the game with a late penalty from flyhalf Manie Libbok that was missed, lead the table with 19 points from second-placed Argentina on 14 points ahead of a final meeting between the sides in Nelspruit next Saturday where the title will be decided.
Argentina scored four tries through wing Mateo Carreras, flanker Pablo Matera, prop Joel Sclavi and Albornoz as they took advantage of a first-half yellow card for visiting wing Kurt-Lee Arendse and turned a 17-0 deficit into a 26-22 halftime lead.
South Africa, who had made 10 changes to their side after resting several regulars, scored tries via fullback Aphelele Fassi, centre Jesse Kriel and scrumhalf Cobus Reinach as they lost for the first time in this year's championship.
South Africa will have to wait another week to claim a fifth Southern Hemisphere crown and need a point from the meeting in Nelspruit to be sure, though Argentina have shown it is far from a foregone conclusion.
"We need to continue applying pressure. There were times in the game when we let go of it," stand-in South Africa captain Salmaan Moerat said.
"That will be the big work-on for us going into next week."
The Springboks demolished Argentina in the first scrum and from the resulting attacking line-out, a simple backline move opened a hole in the home defence and Fassi was able to coast over the tryline.
They scored their second try after more excellent handling that put Kriel into a gap under the posts.
Flyhalf Handre Pollard added a further three points, but when Argentina did finally get the hands on the ball, they scored a superb try.
Some intricate backline play opened up the defence on the left wing and Carreras was able to cross for a converted score.
The momentum changed after that and the Springboks' cause was not helped by a yellow card for a dangerous tackle against Arendse that reduced them to 14 players.
Argentina grabbed their second try in almost the same fashion as their first, opening up the space on the left wing with some excellent hands and Matera was able to cross.
The home side got their third score while Arendse was still off the field as an attacking line-out saw Sclavi barge over and the hosts led 19-17.
The rampant Argentines scored their fourth try as Albornoz found a gap in what was becoming a porous Springboks defence.
The visitors finally found some momentum again just before halftime as Reinach took a quick tap from a penalty and raced over in the corner but they trailed 26-22 at the break.
Pollard and his replacement Libbok landed a penalty each to put the Springboks back into the lead 10 minutes into the second period, but when Albornoz put Argentina back in the lead, they were able to hold on for their third win of the Championship.
