Lille were 2-1 up when Senegal's Sylva left the pitch in the 74th minute with a hand injury that rules him out of Wednesday's Champions League match at AC Milan.
"He's got a fractured finger and will be sidelined for a long time," Lille coach Claude Puel told reporters after the game.
Gregory Malicki, who came on for Sylva, should replace him on Wednesday.
Midfielder Benjamin Gavanon put Nancy ahead with a deflected shot from the edge of the box on 52 minutes but Lille equalised 10 metres later through Mathieu Bodmer before Mathieu Debuchy gave them the lead with a superb free kick on 69 minutes.
Guinea forward Souleymane Youla wrapped it up in added time.
Lille drop a spot to third but stay 14 points behind leaders Olympique Lyon, who won 1-0 at Le Mans on Saturday.
Lyon, trying to become the first club from a major European league to win six consecutive titles, are on 43 points from a maximum 48.
Racing Lens, who beat Girondins Bordeaux 3-0 on Saturday, are now in second place, level on 29 points with Lille and fourth-placed Sochaux, who extended their surprise Ligue 1 run by beating Olympique Marseille 1-0 on Sunday.
A game between Paris St Germain and Toulouse also scheduled for Sunday has been postponed for security reasons.
Police said they feared crowd trouble after a PSG fan was shot dead by a policeman following a 4-2 UEFA Cup defeat by Hapoel Tel Aviv on Nov. 23. No new date has been set yet.
