The Springboks, who had scored nine tries in their 73-13 romp past the same opposition in Johannesburg last weekend, had to rely on Steyn's boot against a much more physical Pumas.
Argentina led for much of the encounter after tries from flanker Juan Manuel Leguizamon and centre Marcelo Bosch, but Pumas indiscipline in the final 10 minutes allowed South Africa to edge ahead and hold on for victory.
Wing Bjorn Basson scored the visitors only try, with Steyn kicking the rest of his side's points to take them to the top of the four-nation tournament's standings on points difference from title holders New Zealand.
The Springboks and All Blacks have nine points after two rounds, with Argentina on a single point and Australia yet to break their duck after two defeats against New Zealand.
It was South Africa's first away victory in the southern hemisphere's elite competition since beating New Zealand in Hamilton in 2009.
South Africa, who were held to a 16-16 draw by Argentina in the same fixture in Mendoza last year, had spoken in the week leading up the test of the improvement they expected from the Pumas after their abject display seven days ago.
But while the home side showed more physicality and passion, South Africa were nowhere near as slick as they made numerous unforced errors.
Leguizamon barged his way over for the opening try inside four minutes as the Pumas made all the early running.
Morne and Pumas captain Felipe Contepomi traded penalties after that, before the Springboks hit back to level the score at 10-10, Basson crossing the line unopposed after the visitors spread the ball wide.
Argentina continued to impress with their physicality and enterprise and had their second try three minutes before halftime when good play from wing Gonzalo Camacho got them close to the Springbok line and outside centre Bosch crashed over, with Contepomi adding his second conversion.
Steyn added a second penalty, but the Springboks still trailed 17-13 at halftime.
The match threatened to boil over in the second period as first Springbok flanker Francois Louw accused an opponent of eye-gouging, before lock Eben Etzebeth claimed he was bitten.
There was no immediate visual evidence of either incident.
The frustration of the visitors was evident, but they closed the gap to a single point with another Steyn penalty five minutes after halftime.
South Africa edged ahead with eight minutes to play as the Pumas were penalised for collapsing a maul.
The Springboks managed to play the remainder of the match in the Argentina half and were rewarded with another penalty when the Pumas again collapsed a maul for Steyn to convert his fifth penalty.
