Yet Milan's defensive frailty was punished when Konstantinos Fortounis scored from the penalty spot in the 81st minute to give his side the win that left them second in Group F on 10 points.
They edged out the seven-times European champions on goal difference and progressed along with group winners Real Betis.
Chelsea dropped points for the first time this season in the competition as they drew 2-2 with Vidi FC in Budapest, but the game was overshadowed by allegations of anti-Semitic chanting by the visiting fans.
Swedish side Malmo pulled off one of the bigger shocks of the round, winning 1-0 away to 10-man Besiktas to finish runners-up behind Genk in Group I.
Besiktas's Ricardo Quaresma was given a straight red card for a reckless tackle on Franz Brorsson in the second half.
Bayer Leverkusen demolished AEK Larnaca 5-1 to finish top of Group A, with FC Zurich finishing runners-up following a 1-1 draw with Ludogorets.
Group C winners Zenit St Petersburg were comfortably beaten 2-0 away by Slavia Prague, who claimed the runners-up spot, rendering Bordeaux's 1-0 win away to FC Copenhagen meaningless as both sides went out.
Celtic, who were beaten 2-1 at home by a Salzburg side that had already secured top spot in Group B, needed a helping hand from Rosenborg.
The Norwegians secured a 1-1 draw at home to RB Leipzig that knocked out the German side and sent the Scottish champions through.
Despite a 3-0 thumping by Sevilla, Krasnodar made it out of Group J as Standard Liege could only manage a scoreless draw away to Akhisar.
They will be joined in next Monday's draw for the knockout stage by Villarreal, Rapid Vienna, Dynamo Kiev and Stade Rennais.
(Reporting by Philip O'Connor; Editing by Toby Davis)
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