The Portugal forward, signed from Porto in the close season, became the first player to score a European hat-trick for the club since Brazilian Kaka against Anderlecht in 2006.
Hakan Calhanoglu set Milan on their way with a fierce drive in the seventh minute and then set up Silva's first two goals in the 10th and 20th of the Group D game.
Although Alexandar Borkovic pulled one back two minutes after the break, Silva completed his treble in the 56th minute after being sent clear by Franck Kessie's through ball. Suso added the fifth just after the hour before Milan eased up.
Milan's fellow Serie A side Atalanta were also impressive, scoring three times in the first half to beat English Premier League opponents Everton in Group E.
Andrea Masiello tapped the ball in from a corner after 27 minutes before Alejandro Gomez rifled home a shot from the edge of the area in the 41st and Bryan Cristante added the third three minutes later after Everton were caught in possession.
Olympique Lyonnais were held 1-1 at Apollon Limassol in the group's other game after Adrian Sardinero Corpa got a last-gasp equaliser for the Cypriots to cancel out Memphis Depay's 53rd minute penalty.
Braga pulled off a shock in Group C by coming from behind to win 2-1 at Hoffenheim, who had beaten German champions Bayern Munich in the Bundesliga on Saturday.
Sandro Wagner gave Hoffenheim the lead in the 24th minute but Joao Carlos Teixeira headed the equaliser on the stroke of halftime from a cross by Ricardo Esgaio who also set up the winner for Dyego Sousa five minutes after the break.
(Writing by Brian Homewood in Berne; Editing by Ken Ferris)
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