The visitors frequently decided to scrummage instead of trying their luck with kicks as coach Eddie O'Sullivan opted to see his team compete with Argentina's fearsome pack ahead this autumn's World Cup.
Pumas fly-half Federico Todeschini showed what could be achieved as he kicked four of five, and made the home fans forget Felipe Contepomi, at least temporarily.
Ireland played their best rugby of the tour in the first half of last week's Test in Santa Fe, and O'Sullivan was left to contemplate another below-par showing on Saturday.
His worries during the week about the indiscipline of his players were borne out during the match.
As in last Saturday's match, Ireland made a series of errors. The players were surprisingly lacklustre, showing imprecision when passing and receiving the ball.
But Argentina struggled to take advantage of the Irish technical mistakes as they centred their game on physicality.
The Ireland backs made some errors during the opening moments of the game and Argentina pushed the visitors back with short passes.
Todeschini's penalty in the sixth minute fired Marcelo Loffreda's side ahead.
Staunton, who replaced Paddy Wallace after he suffered an injury in the last Test, had the opportunity to level the game but he missed after 25 minutes.
On the half-hour mark, Ireland's best move of the game ended after Rob Kearny's forward pass halted the attack just a few metres from the try-line. After that Ireland's nerves increased.
Staunton, a Heineken Cup-winner with Wasps, saw a 27-yard penalty-kick brush the post with two minutes to the interval.
Todeschini then scored again with another penalty in the fourth minute of first half injury-time.
Ireland got better in the second half as they kept hold of possession, but they still could not control their jitters.
After 47 minutes Staunton failed to convert a third penalty.
Todeschini also failed soon afterwards, but he had his revenge when he converted his third and gave Argentina a 9-0 advantage.
Wing Neil Best replaced Alan Quinlan and quickly got into a fight with Francisco Leonelli.
Geordan Murphy dropped the ball poorly and left Argentina close to their goal.
After a five-minute battle in the scrum, Manuel Contepomi avoided the Irish defence and crossed for the only try of the clash in injury-time.
There was still time for Todeschini to wrap up the scoring with another penalty.