First-half tries by Richie Lane, Ciaran Potts and Niall Ronan gave Ireland a 19-12 advantage, and at that stage a bonus-point win and semi-final place looked a real possibility. But in the second half the powerful Argentina pack took over and kept Ireland scoreless.
Other results conspired to put Ireland into second-last place in the group - an unfortunate outcome after they beat Scotland and ran South Africa desperately close. They will play Japan on Wednesday, with the winners taking on either Italy or Canada for 9th place overall.
Argentina now go into the semi-finals. They will play Australia, who ensured qualification with a thrilling 37-37 draw against New Zealand. The Kiwis will face South Africa in the other semi. Elsewhere, France's 21-16 win over England took both teams into the play-offs for 5th-8th, along with Wales and Scotland.
The final round of matches will take place next Sunday.