The result means Argentina's most successful team remains in search of a first away win after also drawing at San Lorenzo and Platense. They are fourth in Group B with nine points from five games while Godoy are 13th on two with a game in hand.
"There are no excuses, it wasn't good, we were very predictable," Marcelo Gallardo, River's manager, said after the game. "We didn't have the aggression to break down Godoy Cruz.
"We lack creativity and also a bit of decision-making. I have good players, it will come."
River pressed forward since beginning while the hosts stayed compact in defence. In crucial final minutes Miguel Borja came close to opening the scoring for River but the Colombian forward missed by just a few centimetres as he tried to beat home side goalkeeper Franco Petroli.