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.
River pressed forward at the beginning while the hosts stayed compact in defence. 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.
Godoy threatened through set pieces later in the first half. Mateo Mendoza came close with a header after a corner routine but his effort went over the crossbar.
River defenders Milton Casco and Fabricio Bustos pushed high up the pitch but could not get a breakthrough while their goalkeeper Franco Armani punched away Nicolas Fernandez's shot to keep the scores level at half-time.
Pity Martinez came close twice in the second half. His volley forced a strong save from Petroli and, minutes later, a powerful left-footed strike was also denied by the goalkeeper. At the other end, Fernandez tested Armani from long range.
River midfielder Manuel Lanzini came on in the second half but was forced off in the 64th minute after sustaining a shoulder sprain. Substitute Marcos Acuna nearly found the opener with a long-range effort that went just wide.
Godoy midfielder Gonzalo Abrego fired from distance and fellow midfielder Luciano Pascual headed from inside the box in the last minutes but Armani saved both to keep the scores even.
River are set to host Lanus on Sunday and Godoy visit Velez Sarsfield on Monday.