Paraguay, who drew their first group match against Qatar, took the lead on 37 minutes when Richard Sanchez struck the opening goal in Belo Horizonte.
Argentina, beaten in their first match against Colombia, started the game with City's Nicolas Otamendi in defence and Sergio Aguero on the bench.
Aguero came on for the start of the second period and created a couple of half-chances before a penalty was awarded by VAR for handball.
Lionel Messi tucked away his 68th international goal from the penalty spot, but minutes later, Paraguay had a golden chance to retake the lead.
Otamendi floored Gonzalez in the box on 62 minutes but the same player then saw the resulting penalty saved by Argentina's Armani.
Paraguay comfortably saw the remainder of the game out to leave Argentina needing a win against Qatar on Sunday to stay in the competition.