Jenkins took charge of Wales six weeks ago after being named as the long-term successor to Mike Ruddock but now knows the size of the task ahead.
Mark Jones and Ian Evans both went over in the first 10 minutes as Wales took an early lead in the first Test to be staged in Welsh-speaking Patagonia.
But Argentina hit back before half-time through Juan Fernandez Lobbe and Jose Nunez Piossek before replacement Francisco Leonelli of Edinburgh sealed the game with the Pumas' third try three minutes from time.
James Hook, who won his first cap off the bench, scored in injury time to cap an encouraging performance from a young Wales team but it was ultimately disappointing after starting so well.
Wales survived an injury scare to wing Mark Jones and outside-half Nick Robinson and their importance was underlined when they combined for the game's opening try inside five minutes.
Robinson scuffed a drop goal attempt that nevertheless resulted in a scrum, and then a penalty that Mike Phillips wasted no time over, passing to his Cardiff half-back partner who found Jones unmarked out wide to score in the corner.
However, it was Phillips' error that allowed the Pumas a swift response from the re-start. The scrum-half saw his clearance charged down by London Irish number eight Juan Miguel Leguizamon who missed with his pass but Lobbe, the younger brother of Sale lock Ignacio, stretched out to touch down.
Federico Todeschini's conversion gave the Pumas a 7-5 lead after seven minutes.
A moment of opportunism saw Wales regain the lead when new cap Evans intercepted Agustin Pichot's pass, as the Pumas pressed for a second, and galloped more than half the length of the field to mark his international debut with a try.
Nick Robinson converted for a 12-7 lead after just 10 minutes but their indiscipline cost them the lead as the Pumas looked to rattle the visitors with a typically physical approach in and around the rucks and mauls.
Irish referee Alain Rolland was forced to speak to both captains on more than one occasion before Wales were shown two yellow cards in quick succession.
Flanker Gavin Thomas, back in the Wales team after a three-year absence, was punished for dissent and then number eight Alix Popham followed to the sin-bin eight minutes later for pulling down a Puma lineout drive.
The Welsh pack was down to six and it was from a strong shove from the Puma scrum that they scored their second try in the 39th minute.
Inevitably Wales conceded 12 points while down to 13 men, Piossek dived over from a powerful scrum and Todeschini added the conversion and two penalties for a 20-12 half-time lead.
Wales then enjoyed the better of the second half but failed to capitalise on breaks by Shane Williams and Mark Jones though two penalties by Robinson brought them to within two points.
However, a scrum against the head deep inside their own half proved costly and Leonelli went over to seal the win for Argentina before Hook's late consolation try.