Warren Gatland's side led three times in the second half but wilted late on, and tries from Northampton prop Martin Scelzo and right-wing Octavio Bartolucci saw Argentina, under new coach Marcelo Loffreda, run out relatively comfortable winners.
Ireland's left-wing Justin Bishop and centre Rob Henderson both scored second half tries as the the lead chopped and changed and Ireland led 18-17 when fly-half David Humphreys - having missed three conversions and a penalty - finally landed a penalty.
Gonzalo Quesada missed an early conversion attempt after full-back Ignacio Carleto had claimed the opening try but then found his range to contribute 19 points with five penalties and two conversions.
Argentina made the early running and might have been lucky to keep their line intact. Malcolm O'Kelly charged down an attempted dropped goal by Quesada before right-wing Bartolucci with a clear run was whistled back for crossing and Ireland were awarded a penalty.
The Pumas broke the deadlock in the 14th minute with a try from full-back Carleto after the the Irish scrum, was turned, and Pumas captain and scrum-half Agustin Pichot took a short pass from number eight Gonzalo Longo before delivering an inch-perfect kick to the corner where Carleto touched down.
But Ireland stunned the Pumas with an O'Kelly try, seven minutes later after a powerful drive from number eight Anthony Foley stretched the home defence.
Quesada' was wide with a drop-goal and penalty and then a mistake by the number 10 gave David Humphreys a straightforward chance to kick the tourists ahead from in front of the posts but he drilled his attempt left and wide.
Quesada landed a penalty from just outside the Irish 22 to give the Pumas an 8-5 lead.
Seven minutes into the second half Ireland regained the lead withh a smartly taken try created by right-wing Shane Horgan and scored by Bishop.
Humphreys was off target with the conversion, but Ireland were ahead for the first time at 10-8.
The lead was shortlived as Quesada landed a penalty to edge the Pumas back in front 11-10.
The tourists stormed back and Henderson burst over for a try in the corner with the home defence in tatters.Hiumphreys again missed the conversion and Ireland were fortunate minutes later when Carleto burst past McKenna but was denied a try when adjudged to have put a foot in touch.
Quesada pulled the Pumas back to within a point with his third penalty and kicked a fourth 10 minutes from time to give the hosts a 17-15 lead.
With seven minutes to go, Humphreys finally broke his duck to kick Ireland ahead again at 18-17, but in the final minutes Ireland wilted and the home side charged home.