Boca succumbed to a 90th-minute breakaway goal for a 3-2 upset defeat by modest Godoy Cruz at their Bombonera home with coach Alfio Basile refusing to speak to reporters and going into crisis talks with manager Carlos Bianchi after the match.
River coach Nestor Gorosito's job is on the line after his team's 1-0 defeat at Arsenal followed a 0-0 home draw with Colon last weekend and early elimination in midweek from the Copa Sudamericana, South America's Europa League equivalent.
Boca, who also went out of the Copa Sudamericana to Argentine champions Velez Sarsfield in midweek, have taken five points and River four after five matches.
South American champions Estudiantes, who beat Huracan 4-1 on Friday, and Rosario Central, held 0-0 by Banfield, are joint top with 13 points each.
Boca made a string of defensive mistakes in Sunday's defeat, conceding an early goal to Godoy Cruz when Matias Jara latched on to a weak back pass from Paraguayan defender Julio Cesar Caceres to skip past goalkeeper Roberto Abbondanzieri and score.
They scored a controversial equaliser in the 36th minute. In a goalmouth melee, the ball appeared to have been cleared off the line by defender Leonardo Sigali but the linesman ruled it had gone in and Boca midfielder Sebastian Bataglia was awarded the goal.
Chile midfielder Gary Medel put Boca ahead with his first goal for the club since his loan move from Universidad Catolica, a diving header after keeper Nelson Ibanez had parried midfielder Federico Insua's fierce drive in the 56th.
Medel, however, was sent off for hitting an opponent in the face with 14 minutes remaining and it got worse for Boca a minute later when Godoy Cruz substitute Martin Aguirre was brought down in the box by defender Claudio Morel.
Midfielder Federico Higuain converted the penalty.
With Boca pressing for a winner, Godoy Cruz hit them on the break in the final minute, Jara's perfect low ball from the left going between defender Hugo Ibarra and Abbondanzieri to substitute Cristian Chavez who tapped in at the far post.
Not even the presence at the Bombonera of one of Boca's most famous fans, U.S. Open tennis champion Juan Martin del Potro, could lift the team from their present gloom.
Arsenal midfielder Cristian Alvarez converted the ninth-minute penalty that gave them victory amid cries from River fans for Gorosito to go.
Promoted Chacarita Junior's Ricardo Zielinski became the first coach to lose his job this season when by mutual agreement he left with the team bottom with no points.
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