The Argentine striker, signed in the off-season from Ajax Amsterdam for just 400,000 euros, bent the ball in to break the deadlock on the stroke of half-time.
Nice have 32 points from 20 matches along with Girondins Bordeaux in fourth and sixth-placed Stade Rennes.
"With our striker in form, we can hope to take points in any game, at home and away," Nice midfielder Fabrice Abriel told French channel beIN Sport.
"We had to start the new year well. We did not think to do that well, it's good because we keep the rhythm."
Nice dominated play but were denied chances by Valenciennes's tight defence until Cvitanich put them deservedly ahead in the 45th minute.
They continued pushing after the interval and were rewarded when Civelli doubled their lead following a free-kick just before the hour.
Eysseric, who signed a contract at the club this week after originally joining on loan, made it 3-0 with a splendid 20-yard strike before Cvitanich wrapped it up with his second goal and a neat pass for Digard.
Ilan volleyed in a Jerome Rothen cross in the 21st minute to put the visitors ahead and Florian Thauvin doubled their lead three minutes into the second half.
Bastia are 13th on 25 points while Reims lie 16th with 19.
- Sports & Recreation
- Renato Civelli
- Ajax Amsterdam
- Valentin Eysseric
- Didier Digard