The visitors should have taken the lead in the 14th minute when Filip Djordjevic got his head on the end of a cross from Issa Cissokho, but the Sochaux keeper pulled off a good save to tip the ball away from the top corner of his goal.
Lucas Deaux was next to threaten for Nantes when he picked out Vincent Bessat, who blazed over the crossbar from just inside the box.
However, it was Sochaux who took the lead against the run of the first-half's play when Stoppila Sunzu headed in from Sebastien Roudet's set-piece delivery.
Alejandro Bedoya almost brought Nantes back into the game after the break, but his headed effort was cleared off the line, while Serge Gakpe also saw a close-range effort kept out by the goalkeeper.
Djordjevic should have equalised for the visitors with 15 minutes left to play, but somehow he headed wide from only a few yards out with the goal gaping in front of him.