Wales took a second-minute lead with Edward Shervington's try, but the Pumas got first-half tries from Francisco Gomez Kodela and Benjamin Urdapilleta.
Benjamin Macome and Federico Meyrelles Torres scored after the break before Will Jones' late Wales consolation after Matias Cortese was sent off.
The highly successful Davey's position was axed by the Welsh Rugby Union as part of a reorganisation of age-grade rugby.
The defeat was Wales' second to Argentina in the tournament, following a 13-10 defeat last week in the pool stages.
Argentina coach Eduardo Viazzo:
"The boys applied our plan to the letter, we were balanced and imposed our speed on the game.
"I think we were good enough to be in the semi-finals but we were let down by crucial individual errors at bad times."
Wales: Craig Evans; Roger Davies, Thomas Riley, Dafydd Lockyer, Aaron Bramwell; Aled Thomas, Martin Roberts; Rhys Gill, Edward Shervington, Craig Mitchell, Christopher Dicomidis, Dominic Day, Will Jones, Rhys Shellard (capt), Adam Powell.
Replacements: Richard Wilkes, Richard Cornock, Craig Everett, Stuart Phillips, Nicholas Griffiths, James Dixon, Christopher Clayton.
Argentina: Gonzalo Sarasqueta; Gonzalo Feijoo, Federico Amelong, Matias Fernandez Gill, Federico Meyrelles Torres; Santiago Fernandez, Francisco Albarracin (capt); Luis Francisco Piccinini, Matias Cortese, Francisco Gomez Kodela, Benjamin Macome, Agustin Smidth, Agustin Creevy, Joaquin Brinnand, Jose Manuel Roan.
Replacements: Juan Avila, Gaston Cortez, Gonzalo Alvarez, Carlos Roldan Verges, Nicolas Bruzzone, Agustin Figuerola, Benjamin Urdapilleta.
Referee: George Clancy (Ireland).