Matteo Gabbia scored with a second-half header at the San Siro then Joao Felix converted a free kick as Milan moved up to seventh place on Saturday.
Earlier, thousands of fans protested outside the club headquarters over Milan failing to qualify for Europe and losing to Bologna in the Italian Cup final.
There were more protests inside the San Siro.
Fans on the upper level at one end positioned themselves to form the words, "Go home." Then they walked out during the early stages of the first half.
There were also chants aimed at Milan's American owner Gerry Cardinale.
"Cardinale, you have to sell. Get out," the fans said.
Milan began the season as potential title contenders but ended up 19 points behind champions Napoli.
The seven-time European champions were eliminated by Feyenoord in the Champions League playoffs in February.
Meanwhile, city rivals Inter Milan will play Paris Saint-Germain in the Champions League final next Saturday.
Genoa won at Bologna 3-1 with two goals from 18-year-old Lorenzo Venturino.
The defeat still didn't prevent Bologna fans from celebrating another strong season, which included an Italian Cup trophy and a Europa League berth.
Bologna were left in ninth place while Genoa was 13th.
Vitinha put Genoa ahead early and Venturino made the most of his first start with two first-half goals, the first of which included some fancy dribbling past several defenders.
Riccardo Orsolini pulled one back for Bologna in the second half.
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