David converted from a Timothy Weah cutback to open the scoring midway through the first half and he then sent the goalkeeper the wrong way from the penalty spot in the 57th minute after United States international Weah had been brought down in the box.
Ibrahima Sissoko pulled a goal back for Strasbourg but they then had Adrien Thomasson sent off as they finished the game in Alsace with 10 men.
It is a second straight win for reigning champions Lille, whose title defence got off to a poor start.
They are back up to eighth, three points behind Marseille in second albeit having played two games more.
Lille go to Austria to play Red Bull Salzburg in the Champions League in midweek.
David, who was born in New York, scored 13 goals for Lille last season after signing from Belgian club Gent.
His team won all 11 matches in which he found the net as they pipped Paris Saint-Germain to the title.
Their three wins so far in this campaign have come when the 21-year-old has been on target.
Meanwhile Nice moved provisionally up to third with a 3-0 defeat of winless Saint-Etienne, who are now bottom of the table.
Amine Gouiri gave Nice an early lead from a deflected free-kick before Calvin Stengs and Andy Delort added further goals in the second half.
Leaders PSG, who are seven points ahead of Marseille in second, are looking for their eighth win in as many outings this season when they host Montpellier later.
Mauricio Pochettino's side are again without Lionel Messi, who misses a second straight game with a knee problem.
