Nene, PSG's best player this season, angled a shot into an empty net after snatching the ball from Baba Tchagouni when the goalkeeper fluffed a routine clearance in the 14th minute.
The Brazilian came close to adding another before the interval but his low shot from just outside the box shaved the post as the visitors piled on the pressure.
Dijon, however, reacted after the break, with Brice Jovial being denied by Nicolas Douchez 20 minutes from time.
Ligue 1 leaders PSG, who take on second-placed Montpellier on Sunday, had midfielder Mathieu Bodmer replaced by striker Kevin Gameiro due to a possible thigh injury with 15 minutes remaining.
Earlier, Olympique Marseille secured their place in the last eight when Brandao inspired them to a 3-1 win over fourth division Bourg Peronnas.
Brandao, who has scored five goals in six games since coming back last month from Brazil, latched on to a cross by Cesar Azpilicueta on 28 minutes to open the scoring.
The Brazilian striker doubled the tally on the stroke of halftime after being perfectly set up by Morgan Amalfitano.
Jordan Ayew, back from the Africa Cup of Nations where his Ghana side reached the semi-finals, made it three with a low shot nine minutes into the second half.
The amateurs scored a consolation goal through Ousmane Diaby 12 minutes from time.
Bourg Peronnas were supposed to host Marseille at Geoffroy Guichard but risks of snow falls in Saint-Etienne led the French federation to swap stadiums.
