Seven-time European champions Milan, who have never won Europe's second-tier competition, lost 2-0 after going behind in the 16th minute when Dodi Lukébakio cut inside and fired into the bottom corner from 20 metres.
The Lisbon side were 2-0 up early in the second period through Jakub Kaminski.
Benfica, twice European champions in the 1960s who have also never won the UEFA Cup/Europa League, hit the post too as they easily got the better of fellow Portuguese Ruben Amorim and his heavily rotated Milan side.
Milan's Portugal striker Gonçalo Ramos, who began his career at Benfica, was largely ineffectual at the San Siro.
Leverkusen have endured an up-and-down start to the Bundesliga season but were boosted on 15 minutes when Svit Seslar deflected Alexi Garcia's cross into his own goal.
The Werkself are the top-ranked team in the Europa League based on UEFA co-efficients and did enough to see off the Slovenians, with Facundo Medina volleying in a corner in the 74th minute.
Lyon won 2-1 at Anderlecht and Sunderland, in their first European action since 1973, edged AZ Alkmaar 1-0 in England's North East thanks to Enzo Le Fée's penalty.
Le Fee had won the spot-kick when he was fouled by Weslley Patati as he attempted a shot.
Sunderland then rode their luck as Alkmaar missed several superb chances.
Olympiacos came from behind to beat Poland's Jagiellonia Bialystok 2-1 in Greece, Sturm Graz were held 0-0 by Rennes and Israel's Hapoel Beer-Sheva also drew 0-0 with Dinamo Zagreb in a game played in Bucharest.
Ararat-Armenia, the first Armenian club to qualify for the Europa League proper, earlier lost 4-1 at home to Sparta Prague.
Norwegian Tobias Guddal scored twice in quick succession for Sparta midway through the first half in Yerevan. First he headed in a corner and then rifled home after the hosts had failed to deal with another corner.
John Mercado and Patrik Vydra added to the Czech side's score after the break, but there was something to cheer for home fans when Bruno Wilson pulled a goal back.
In the other early kick-off, Omonia Nicosia beat Spaniards Celta Vigo 1-0 in Cyprus.
The nine remaining games on matchday 1 take place on Thursday, including Juventus v Dutch side NEC, Real Sociedad taking on Bournemouth and OFI Crete meeting Hoffenheim.
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