Hosts Rangers paid the price for a harsh second yellow card issued to defender Jefte as Dynamo struck twice in the final 10 minutes through substitutes Oleksandr Pikhalonok and Nazar Voloshyn.
The Ukrainian side advanced 3-1 on aggregate and head into a play-off tie against Salzburg, who edged FC Twente 5-4 after a 3-3 draw in the return leg in the Netherlands.
Jonathan David's penalty secured Lille a place in the next round as the French club snatched a 3-2 aggregate victory in Turkey, marking a disappointing start to Mourinho's time in charge of Fenerbahce.
Fenerbahce lost the first leg 2-1 last week and appeared 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 Lille held on. Their reward is a meeting with Slavia Prague, who were 4-1 aggregate winners over Union Saint-Gilloise.
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.
Sweden's Malmo won a thrilling tie with PAOK, striking late in Greece to force extra time and then winning the second leg 4-3 to prevail 6-5 overall.
Sparta Prague beat Romanians FCSB 4-3 on aggregate, while Qarabag stunned Ludogorets with four goals in extra time to pull off a remarkable 7-2 win in Bulgaria.
Switzerland international Kwadwo Duah netted twice on the night to put Ludogorets 4-2 ahead in the tie, but the visitors from Azerbaijan scored six times without reply to run away with an 8-4 aggregate success.
