River's traditional rivals Boca Juniors were also beaten, suffering a shock defeat in Ecuador against Deportivo Cuenca, although the six-times former champions should still qualify for the last sixteen.
River's third straight away defeat left them with only four points from five games and meant Peruvian champions Deportivo San Martin qualified from Group Three along with Uruguay's Nacional.
Colombian striker Radamel Falcao Garcia gave River the lead in the 43rd minute in Asuncion when he converted a penalty after a foul on striking partner Cristian Fabbiani.
But the hosts levelled in first half stoppage time when Fabio Escobar scored, also from a penalty.
River fell apart after Diego Buonanotte was sent off for a bad tackle in the 58th minute.
In a 13-minute spell, substitute Jorge Nunez scored twice for the Paraguayans and Carlos Ruiz was also on target as they raced to a 4-1 lead.
Marcelo Gallardo made the score more respectable when he pulled one back with nine minutes left.
Boca, missing playmaker Juan Roman Riquelme and key defender Julio Cesar Caceres, stayed top of Group Two with 12 points despite losing their 100 percent record in Cuenca.
Substitute Edison Preciado scored the only goal with a delightful finish in the 78th minute, one of only a handful of scoring chances in the match.
Cuenca have nine points, level with Venezuelan champions Deportivo Tachira. Boca still need a draw from their final match at home to Tachira to qualify.
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