Tries by wing Liam Williams and scrumhalf Gareth Davies, who both missed the Australia match, were cancelled out by touchdowns for Pumas centre Juan Martin Hernandez and scrumhalf Martin Landajo, with 14 points from the boot of Leigh Halfpenny giving the home side the advantage.
“We knew we were pretty poor in the first game (against Australia) and we had to play and be a credit for the people who turned up today," said lock Alun Wyn Jones.
"We've taken a lot of criticism, rightly so, but we wanted to prove people wrong," added the man of the match after Wales ended a run of five test defeats.
Wales should have led by more than 6-3 at halftime after camping on the edge of the Argentina line for the last quarter of an hour.
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But despite a succession of penalties, including one which earned prop Ramiro Herrera a yellow card, Wales failed to turn 63 percent possession into points.
After the restart, Wales soon found themselves back at the Argentine line and a tight TMO decision gave Williams the try following an attack on the right.
Argentina hit back quickly from a tap penalty when Landajo kicked a grubber which Hernandez chased to touch down by the posts. The conversion from flyhalf Nicolas Sanchez took the Pumas to within a point at 11-10.
A rare kick ahead by prop Gethin Jenkins, captain in a Welsh record 128th test, gave the home side a lineout close to the Pumas line and Davies went over from the maul with Halfpenny converting.
Once again, the Pumas hit back with Landajo being awarded a try in another tight TMO decision after going over from a maul.
With the Pumas trailing by a point, Wales played safe with Halfpenny kicking a 78th-minute penalty.
Next Saturday, Wales host Japan, who were beaten 54-20 by the Pumas in Tokyo last weekend but won 28-22 in Georgia on Saturday, while Argentina visit Scotland.
(Reporting by Rex Gowar, editing by Neil Robinson)
