They scored the first try in the opening minutes when Richard Lane took a nice line on Michael Rainey's pass, and five minutes later they scored their second, this time Ciaran Potts muscling his way over from a line-out close to the Argentine line.
The Argentines came back with a try from Francisco Bosch but this didn't phase the Irish who responded with a try from the excellent Niall Ronan, a score that was converted by Rainey to leave the Irish in front, 19-5.
From then on things began to go a little pear-shaped. Argentina scored a try through their prop Euseaid Guiveau and the conversion left just a score (19-12) separating the sides at the break.
The Irish were rocked by an early second half try from Marcello Torres and Cordone gave them the lead with a penalty in the 50th minute.
After that the Argentines used their strength up front and pace in the backs to pin the Irish down and they went further ahead with a try three minutes before the end of normal time, scoring another deep in injury time to secure for themselves a place in Wednesday's semi-final.
Ireland. Peter O'Brien (Tommy Bowe 53); Richard Lane, Keith Matthews, Glen Telford, John Hearty; Michael Rainey (capt, David McAllister 67), Fiach O'Loughlin (Conor O'Loughlin 20); Jason Moran (Eugene McGovern 53), Rory Best, Declan Fitzpatrick; David Gannon, Neil McComb (Simon Crawford 67); John Muldoon, Niall Ronan, Ciaran Potts.
Argentina. Francisco Bosch; Tomas De Vedia, Juan Ignacio Gauthier, Mariano Lecot (Horacio San Martin 72), Leopoldo Villaneuva; Alfredo Cordone, Marcelo Torres (Alfredo Lalanne 77); Eusebio Guinazu, Alberto Vernet Basualdo, Joaquin Gonzalez (Pablo Henn 69); Esteban Lozada, Pedro Zabaloy; Andres Bordoy (Juan Manuel Leguizamon 55), Alvaro Galindo, Jose Ignacio Ghidara (Facundo Barni 77).