Both sides pushed for the game's crucial third goal before half time, with Emmanuel Mayuka, Rafael Dias and Cedric Bakambu coming close for the hosts, while Radamel Falcao and Mounir Obbadi threatened for Monaco.
It was more of the same at the start of second half as Rodriguez was denied from close range by Pouplin moments before Mayuka rattled the bar at the other end.
Sochaux came even closer to pulling a goal back in the 54th minute when Dias's corner was met by Sebastian Roudet, but the midfielder's effort was cleared off the line.
They wouldn't have to wait much longer for a goal, however, as another Dias corner three minutes later found Bakambu, who rose highest to nod the ball beyond Danijel Subasic.
It was Dias yet again who provided the equaliser with another fine delivery from a corner, this time finding the head of Joseph Lopy, who made no mistake to level the sides.
Neither side could find a winner in the closing stages as Sochaux lifted themselves off the bottom of the table, while Monaco now trail Paris Saint-Germain by two points at the top.