Fenerbahce lost the first leg 2-1 last week and looked to be going out until Bafode Diakite's late own goal sent the match to extra time.
Lille substitute Aissa Mandi was sent off on his debut for a reckless challenge on Irfan Can Kahveci, but the visitors were awarded a penalty for a handball by Jayden Oosterwolde and David made no mistake from the spot.
Cenk Tosun headed against the bar as the French club held on to advance to the final qualifying round, where they will meet Slavia Prague after they defeated Union Saint-Gilloise 4-1 over two legs.
Fenerbahce's exit marks a disappointing start to Mourinho's time in charge as he looks to lead the club to a first Turkish title since 2014.
Slovan Bratislava, who entered in the first qualifying round, overcame APOEL Nicosia 2-0 over two legs.
They will take on Midtjylland for a place in the new-look, 36-team tournament after the Danish champions beat Hungary's Ferencvaros 3-1 on aggregate.
Norwegian champions Bodo/Glimt eased through with a 4-1 win over Poland's Jagiellonia Bialystok, completing a 5-1 aggregate triumph.
Salzburg edged FC Twente 5-4 after a 3-3 draw in the return leg in the Netherlands.
Elsewhere, 10-man Rangers lost 2-0 to Dynamo Kyiv at home after a 1-1 draw in Poland last week, meaning the Glaswegians depart the competition along with FCSB, who were edged out by Sparta Prague 4-3 on aggregate.
In the craziest clash of the lot, Qarabag defeated Ludogorets 7-2. After forcing extra time with a lead of 3-2, Qarabag added four extra-time goals to secure progression to the final qualifying round.