The touchdown capped a late fightback by the Springboks who trailed by 10 points with 10 minutes left.
Number eight Whiteley went over to spare their blushes but there will be a slew of questions about the chances of the South Africans in the annual southern hemisphere championship after World Cup winners New Zealand crushed Australia earlier on Saturday.
The Boks scored three tries to two for Argentina, who will consider themselves desperately unlucky not to win in South Africa for a second successive year.
"We need to learn how to close out a match," said Pumas captain Agustin Creevy on his 50th appearance. "We are really disappointed we were not able to defend our lead but we've improved a lot over the last year."
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It was the home side who started brightest with a seventh-minute try as a break by flyhalf Elton Jantjies set up winger Ruan Combrinck in the right corner.
After trading penalties, South Africa had a chance to extend their seven-point lead when Manuel Montero was yellow-carded for lifting an opponent in the tackle and Argentina went down to 14 men.
However, Nicolas Sanchez kicked a penalty for the Pumas before centre Matias Orlando dived over at the end of a sweeping move.
South Africa should have scored a second try when centre Lionel Mapoe crossed the line on the stroke of halftime but he let the ball slip from his grasp.
SANCHEZ LANDMARK
Jantjies' penalty 15 minutes into the second half levelled matters at 13-13 but Sanchez, passing 400 test points in the process, restored Argentina's lead with a quarter of an hour to play with a penalty of his own.
A mistake then allowed Montero to break down the wing to set up a try for Santiago Cordero as Argentina went 23-13 up.
But the Boks grabbed their second try within a minute as a strong scrum and a darting break by scrumhalf Faf de Klerk and a switch pass allowed Johan Goosen to dive over before a conversion by Jantjies brought them back within three points.
With the crowd suddenly lifted, pressure play from South Africa in the last 10 minutes forced a penalty from easy range and Jantjies slotted it over to equalise.
Whiteley then clinched victory to set up an intriguing encounter when the two teams meet again next Saturday.
"We showed a lot of pride and the crowd definitely pulled us through. They gave us power but it was a game in which we struggled to hold on to the ball," said De Klerk, the catalyst for South Africa's comeback.
(Reporting by Mark Gleeson; Editing by Ed Osmond)
