The Rossoneri brought momentum into this match after an emphatic win over Real Madrid on matchday four, and were in immediate control here.
Yet it was the hosts' David Strelec who created the first big chance after 15 minutes, rounding Mike Maignan, but his effort was spectacularly cleared off the line by Strahinja Pavlović.
That wake-up call seemed to energize the visitors as Christian Pulisic soon took center stage, forcing Dominik Takáč into a sharp save near the post before breaking the deadlock with a clinical low drive into the bottom corner after a counter-attack.
Vladimír Weiss' men responded quickly, however, as Tigran Barseghyan sliced through a disorganized Rossoneri backline before lifting a sublime finish over Maignan to restore parity in stunning fashion.
One of only five teams yet to score a single point in the UCL competition stage, the equalizer lifted the hosts' spirits, leaving Milan struggling to regain their early control after the break.
Paulo Fonseca tried to inject new energy into his attack after the break by introducing Rafael Leão in place of Noah Okafor. And while Slovan were initially stubborn at the back, a porous defense that conceded four times in each of their previous two Champions League home defeats eventually faltered.
At half-time, Leão restored Milan's lead, grabbing Youssouf Fofana's defensive pass and firing a point shot past Takáč, before Tammy Abraham received a gift from Strelec's misplaced pass to score his side's third just three minutes later yielded.
From then on, Slovan appeared for the count, but in the 88th minute Nino Marcelli struck a blistering left-footed strike into the top corner to raise hopes of a late comeback.
However, that never materialized, as Marko Tolić was even sent off late for a pair of yellow cards in quick succession, while the home side remained bottom of the Champions League table.
Meanwhile, Fonseca's men recorded a third straight European victory, a feat they last achieved in February 2023, putting them within touching distance of the automatic qualifying places.
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