River dominated most of the match and striker David Trezeguet was just wide with two good chances but they were let down by poor defending at a set piece, a regular weakness in coach Matias Almeyda's team.
With 20 minutes remaining, Racing won a free kick on the right and left back Matias Cahais ghosted into the middle unmarked to head the ball into the net from close to the penalty spot.
Before the goal, Uruguayan substitute Rodrigo Mora headed River's best chance with goalkeeper Sebastian Saja making a superb diving save to deny the home side.
No matter that the season does not end until June - relegation quickly becomes a serious issue for many teams due to the points averages in the league's convoluted system.
After two successive defeats and five matches without victory, River's average of 1.125 points from eight matches leaves them one from the bottom of the relegation table.
River, one of Argentina's 'Big Five' clubs with a record 33 league titles, were relegated for the first time in June 2011 when they lost a playoff against then-second division side Belgrano, prompting rioting from fans.
Diaz steered River to five league titles between 1996 and 2002 but he and club president Daniel Passarella do not get on and Passarella publicly gave Almeyda his backing last week when media were casting doubt on the coach's future.
"We've got to press on, work is the only word that fits this situation. I have faith in my work and the squad. We'll pull through," Almeyda told reporters.
