Playing at home in a packed Monumental, River managed the solitary score after half-time when striker Mariano Pavone volleyed home Ariel Ortega's corner.
They are three points adrift of leaders Velez Sarsfield, who beat lowly Olimpo 3-0 on Friday, and two behind Estudiantes with 14 points.
Quilmes, the only team without a win after eight rounds of the Apertura championship, equalised in the second minute of added-time when midfielder Miguel Caneo headed home a free-kick at the far post.
Estudiantes moved within sight of top place with a 1-0 home win over Boca Juniors on Saturday, stopping striker Martin Palermo's team in their tracks after they had won their previous two matches.
Juan Sebastian Veron's side can go top if they win their game in hand, the postponed La Plata derby against arch-rivals Gimnasia, who are one from bottom with five points, on Wednesday.
Veron, who had a brilliant match at the heart of his team, said Estudiantes had put behind them the disappointment of elimination in midweek from the Copa Sudamericana, South America's equivalent of the Europa League.
"Defeats bring criticism, doubts and squabbles... We needed a match like this," he told reporters.
Independiente got their first win, 1-0 at home to Gimnasia, while they attempt to sign a new coach after Monday's departure of Daniel Garnero and the refusal by Americo Gallego, who did not have his contract renewed in June, to return to the club.
The remaining unbeaten records fell at the weekend with Argentinos Juniors beating Banfield 1-0 and Colon's 2-0 home win over previous leaders San Lorenzo, who are now third with 15 points.
