David pounced to tap home in the first minute of stoppage time after Xeka's powerful low drive was spilled by Reims goalkeeper Predrag Rajkovic to put Lille on 42 points, equal with champions PSG and two points ahead of Lyon in third.
Jonathan Bamba's long-range leveller two minutes after the break gave the hosts a chance to snatch the win after Arber Zeneli had given Reims a 35th-minute lead with a cross that bamboozled home stopper Mike Maignan.
Rudi Garcia's Lyon take on mid-table outfit Metz at the Groupama Stadium in Sunday's late match knowing a win would allow them to reclaim top spot.
Lyon are unbeaten in 16 Ligue 1 games, their best run since 2005 and the longest current run of any team in Europe's so-called "Big Five" leagues.
Monaco are six points off the lead in fourth after Wissam Ben Yedder's brace gave them a 3-2 win over Montpellier on Friday.