The Istanbul side dominated to beat Anderlecht 3-0 on Thursday and take a comprehensive lead to Belgium for the second leg next week.
Two of Mourinho's former clubs, Porto and Roma, drew 1-1 later in Europe's second-tier competition.
Fenerbahce only just reached the playoffs by placing 24th in the new format - the last qualification place, and on a tiebreaker of goal difference.
They took the lead in the 11th minute after Anderlecht failed to clear a corner and the ball was put back to the area where Yusuf Akcicek nodded it on for Dusan Tadic to head into the net.
Edin Dzeko then doubled his team's advantage shortly before halftime following good work from Filip Kostic down the left flank.
Dzeko's initial shot was blocked by Anderlecht defender Lucas Hey but the veteran Bosnian forward fired home the rebound.
Youssef En-Nesyri added a third with a powerful header in the 57th minute and Fenerbahce missed several chances to extend their advantage.
Notably, Yusuf Akcicek's header was deflected onto the post - it would have been the 19-year-old's first career goal, after recording his first-ever assist in the first half.
Elsewhere, Ajax won 2-0 at Union Saint-Gilloise, with 16-year-old Jorthy Mokio netting his side's second goal.
Mokio scored in the 71st minute. A corner was cleared but only as far as the Belgium Under-21 international on the edge of the area and he volleyed it in for his first senior goal for Ajax.
Mokio, aged 16 years and 350 days, became the youngest player to score in a knockout stage match in the Europa League, after Christian Rasmussen had netted the opener just before the hour.
Real Sociedad won 2-1 at Midtjylland, while Ferencvaros beat Viktoria Plzen 1-0.
AZ Alkmaar are close to the last 16 after a big 4-1 win against 10-man Galatasaray. Roland Sallai cancelled out Sven Mijnans opener, but Troy Parrott restored the home side's lead in the 37th.
Kaan Ayhan was sent off in the 51st and Alkmaar never looked back, with Jordy Clasie and David Møller Wolfe completing the win.
FCSB came from behind to defeat PAOK 2-1 and Twente scored a penalty in added time to defeat Bodø/Glimt 2-1.
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