River went ahead twice in the match only to be pegged back and were lucky not to concede a third goal to Godoy Cruz in the dying minutes when Cesar Carranza shot wide from a good position.
In a brilliant counter-attack in added time, former Argentina forward Ariel Ortega played a superb ball from the right flank to substitute Funes Mori who chested the ball down on the run and rifled a shot into the top corner.
The linesman, who was up with the action, did not wave his flag but referee Federico Beligoy, close to the half-way line, ruled out the goal for handball. Video replays showed Funes Mori had chested the ball down without using his shoulder.
The 2-2 result left Godoy Cruz and River equal on 17 points from 11 matches.
They are nine points behind Apertura championship leaders Estudiantes, who came from behind to draw 1-1 at Colon.
Earlier on Sunday, Racing Club came from a goal down to beat title holders Argentinos Juniors 2-1 with both goals by Colombian playmaker Giovanni Moreno.
For the first time in the championship, Boca Juniors did not concede a goal as they beat Huracan 2-0 at home with goals from strikers Martin Palermo and Lucas Viatri.
Velez Sarsfield and Arsenal, joint second, won their matches to cut Estudiantes' lead to three points. Velez beat Quilmes 2-0 on Friday and Arsenal won 3-2 at Gimnasia in La Plata on Saturday.
Quilmes' former Argentina youth team trainer Hugo Tocalli became the fifth coaching casualty of the season when he resigned after the match with the winless team on six points.
Velez strikers Juan Manuel Martinez and Uruguayan Santiago Silva lead the scorers list on seven goals while Palermo and Godoy Cruz midfielder David Ramirez have six.
