Defender Julio Barraza and midfielder Rodrigo Gomez scored the goals in the final 10 minutes of an intense Clausura championship clash in which River were looking to join Newell's Old Boys at the top of the standings.
Instead, Newell's can go six points clear if they win at All Boys in the capital on Monday night. Newell's have 32 points with four matches remaining, while River and Lanus, who were held 0-0 by Belgrano, both have 29 with three left to play.
Argentinos emerged from a traumatic build-up to the match with a lot of credit, securing their first win in nine matches despite the club removing three veterans from the squad for not pulling their weight.
One of them, former Argentina defender Diego Placente, was involved in a media war of words with coach Ricardo Caruso Lombardi, accusing him of dropping the trio without notifying them personally.
"Sometimes you've got to cross the line to wake people up," the outspoken Caruso told Futbol Para Todos on the pitch after the victory.
"But we have three more battles ahead," he added.
The relegation dogfight also involves big guns Independiente, who were held 0-0 at home by Estudiantes on Saturday, and Quilmes, who beat Racing Club 1-0 on Friday.
The teams in the bottom three places of the relegation standings, which are based on average points over three seasons, are Independiente, San Martin and relegated Union.
Immediately outside the relegation places are Argentinos, who are five points better off than Independiente.
Independiente visit River Plate next weekend and cannot afford to drop more points, while Argentinos visit San Lorenzo, who have slim title hopes.
Union gave themselves a boost by winning the last Santa Fe derby before playing in the second-tier Primera B Nacional next season, beating bitter rivals Colon 1-0 in a match played behind closed doors for security reasons.
In River's first defeat in 10 matches, Barraza put Argentinos ahead when the ball came to him unmarked at the far post from a corner on the left and he fired past goalkeeper Marcelo Barovero.
With River, who had a Manuel Lanzini strike disallowed for offside and two Rodrigo Funes Mori chances saved by Nereo Fernandez, looking for an equalizer, Pablo Hernandez passed square from the right for Gomez to score and secure the points.