David scored eight minutes into added time with a well-taken shot into the top corner amid whistles and boos from the home fans. It was his 11th goal in 16 appearances in all competitions.
Mohamed Bayo added the second on the break three minutes later to give Lille more lustre to a scrappy win.
Lens, who looked solid throughout but cracked at the very end, slumped to their first league loss.
Tempers frayed at the end as players and staff from both sides got involved in heated exchanges of words and gestures.
Lille was awarded the late penalty after a VAR check which confirmed that Kevin Danso handled the ball in the area.
"You can feel that strength of character," Lille defender Bafodé Diakité said. "They posed us problems, but our strength at the moment is to play all together, to try not to concede goals and not giving up until the end."
David celebrated his goal by mimicking a machine gun with his hands, and received a yellow card.
Lille's second straight road win followed a 3-1 victory at Atletico in the Champions League. After scoring the winner against Real Madrid this month, David had a pair of goals against Atletico on Wednesday.
Lille stayed in fourth place, extending their lead over Lens to three points.
Angers and Saint-Etienne are struggling for goals but turned on a lively contest in which Angers won 4-2.
Ibrahima Niane put hosts Angers back in front from the penalty spot in the 69th minute and Bamba Dieng completed their first win in nine league matches in added time.
"We waited for this win for so long," said Jean-Eudes Aholou, who also scored for Angers. "We suffered a lot to get it."
Angers moved to 14th overall, level on points with Saint-Etienne but with a better goal difference.
Zuriko Davitashvili scored twice. The highly-rated winger joined from Bordeaux this summer as Saint-Etienne prepared to return to the top-flight following a two-year absence.
Brest scored twice within the opening 18 minutes and held on to win 2-1 against Reims.
Unbeaten in the Champions League, Brest have not been so successful domestically, losing four matches already. Saturday's result moved them into eighth place, one point behind Reims.
Leaders Paris Saint-Germain travel to Marseille on Sunday for the latest installment of Le Classique, the biggest game in French soccer.
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