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A quiet first half saw Wales take a 9-6 lead at the break.
Argentina snatched the lead in the 56th minute with a try from Juan Imhoff.
The Pumas scored a second try less than 10 minutes later to open an 11-point advantage.
Wales had the chance to get a try back with less than two minutes remaining but were unable to complete the move.
Wales face Samoa in their second match of the series on Friday.