The lead changed hands three times in an exchange of penalties between Pollard and Pumas flyhalf Nicolas Sanchez before the Springboks edged ahead.
Pollard's first try, just after the halftime siren, gave his side a 24-13 advantage at the break.For much of the opening half it seemed the Springboks might squander their huge volume of possession, just as they did in the 16-16 draw with New Zealand in their previous match.
They struggled at first to use the ball effectively in outside channels, but that changed in the second half as they played with more confidence and combination while Argentina visibly tired.
After Pollard scored his second try in the 53rd minute, outstanding handling and passing produced tries by wingers Makazole Mapimpi and Cheslin Kolbe.
Australia beat New Zealand 47-26 earlier on Saturday but it was South Africa who won the title for the first time, 10 years after winning its predecessor, the Tri-Nations Championship."It's a fantastic feeling," captain Duane Vermuelen said.
"It's a nice thing for the team and also a good thing for us going into a World Cup."We play the best teams in the southern hemisphere so it's good to have this step up. There's still a lot of work to be done but it's a good feeling."