Despite the loss, Argentina remain atop the 10-team round-robin standings with 22 points from 11 matches. Brazil, meanwhile, are temporarily in third place with 17 points.
In a tense encounter in Asunción, Argentina took an early lead in the 11th minute through Lautaro Martinez, whose precise shot was upheld after a VAR review.
Paraguay responded brilliantly, with Antonio Sanabria executing a stunning bicycle kick to equalise in the 19th minute.
Before halftime, Paraguayan defender Gustavo Gómez nearly turned the game with a powerful header that rattled the bar.
The decisive moment came just after the break when Omar Alderete headed in Paraguay’s winner in the 47th minute.
Messi, heavily marked throughout, expressed frustration at referee Anderson Daronco for what he perceived as leniency towards Alderete’s physical challenges.
Brazil’s encounter in Venezuela was no less dramatic.
Raphinha gave the Seleção a 43rd-minute lead with a sublime free kick, but Venezuela’s halftime substitute Telasco Segovia equalised just after the restart with a thunderous effort from distance.
Vinicius Junior, Brazil’s standout forward, had an opportunity to seal the game when he won a penalty in the 67th minute. However, his tame effort was saved by Rafael Romo, and his follow-up missed the target.
Elsewhere, Ecuador cruised to a 4-0 victory over Bolivia, with goals from Enner Valencia, Gonzalo Plata (two), and Alan Minda. The result leaves Ecuador in fifth with 16 points, keeping their qualification hopes alive.
Friday’s qualifiers see Colombia face Uruguay, while Peru hosts Chile in a clash between the bottom two teams.