The home side almost doubled their lead a minute later, but Sloan Privat couldn't find the finish.
Grzegorz Krychowiak drove forward for Reims on 27 minutes, but his effort was saved well by the goalkeeper.
The home side increased the pressure on their visitors in the second half and should have doubled their lead, but Dias could only find the post with his effort.
Dias was at the heart of the action again and was presented with a great opportunity on 70 minutes, but fired wide of the post.
Reims began to push forward in search of an equaliser, though Sochaux stood firm and secured all three points.