Gonçalo Ramos added a late fourth goal to complete an impressive away performance for Rúben Amorim’s side, who responded emphatically after losing their opening league-phase match against Benfica.
The result gave Milan their first three points of the European campaign and lifted their confidence ahead of another demanding run of fixtures.
Cisse opens the scoring for Milan
Milan gradually took control after Salzburg threatened during the opening exchanges, with Adrien Rabiot firing wide before Cisse tested goalkeeper Alexander Zawieschitzky.
The breakthrough arrived in the 15th minute when Omari Hutchinson delivered an accurate cross for Cisse, who rose to head the ball into the net and give the visitors a 1-0 lead.
Salzburg attempted to respond, but Milan remained dangerous on the counterattack and continued to create opportunities through their attacking players.
The Rossoneri doubled their advantage in the 32nd minute following an excellent team move. Rabiot initiated the attack before Davide Bartesaghi supplied Cisse, who finished with his left foot to register his second goal of the evening.
Milan went into half-time with a valuable two-goal cushion after a clinical first-half display.
Cisse completes a remarkable European hat-trick
Milan wasted little time extending their advantage after the restart.
In the 49th minute, Rabiot delivered another inviting cross, and Cisse completed his hat-trick with a powerful header to make it 3-0.
The performance marked a significant milestone for the 19-year-old, who became the youngest Italian player to score a hat-trick in European club competition.
His three goals also made him the first Milan player to score a hat-trick in a continental competition since Filippo Inzaghi achieved the feat in 2002.
Cisse’s clinical finishing gave Milan complete control of the match and left Salzburg facing an increasingly difficult task.
Ramos adds a fourth as Milan finish strongly
Salzburg struggled to find a route back into the contest as Milan continued to create chances.
Luka Modrić produced an excellent pass to release Hutchinson, whose subsequent effort narrowly missed the target.
The visitors added a fourth goal in the 72nd minute when Ardon Jashari won possession and supplied Ramos, who finished with his left foot to make it 4-0.
Milan’s evening was not entirely without complications, however. Filippo Terracciano was sent off in the 83rd minute for a professional foul that denied an opponent a goalscoring opportunity.
Despite finishing the match with 10 men, the Italian side comfortably protected its advantage and secured a clean sheet.
Milan revive their Europa League campaign
The victory was an important response from Amorim’s team after their 2-0 defeat to Benfica in the opening round of the league phase.
Cisse emerged as the standout performer, while Rabiot’s creativity and Hutchinson’s contribution in attack helped Milan produce a convincing team display.
Milan will now turn their attention to domestic action before travelling to England to face Bournemouth in their next Europa League fixture on Thursday, 22 October.
For Salzburg, the heavy home defeat was a disappointing setback, while Milan will hope their performance in Austria marks the beginning of a sustained improvement in Europe.