Eager to make amends following their matchday one defeat to Benfica, the visitors certainly had the better of the opening stages of the game, which saw early opportunities for Adrien Rabiot and Omari Hutchinson go to waste.
They took a deserved lead, though, with a quarter of an hour played, after Cisse headed home Hutchinson’s excellent cross from the right flank.
Milan’s dominance was almost rewarded with a second goal just two minutes afterward, when Christian Zawieschitzky did well to block Ruben Loftus-Cheek’s close-range strike, before a shot from Hutchinson was superbly blocked by Kevin Boma just in front of the goal line.
The hosts had the ball in the back of the net at the midway point of the first half through Haris Tabakovic, but the goal was instantly ruled out due to the ball going out of play during the build-up.
The Rossoneri continued to apply pressure on the Salzburg defence, and Milan doubled their advantage after 32 minutes through Cisse with a lovely improvised finish following Davide Bartesaghi’s pass across the face of goal.
The Rossoneri were in no mood to rest on their laurels in the second half, and it was Cisse who extended Milan’s lead just four minutes after the interval to seal his hat-trick, after he met Rabiot’s pass to slot the ball into the net from inside the six-yard box.
With the result beyond doubt, both Hutchinson and substitute Christian Pulisic came close to adding a fourth midway through the second half, before fellow substitute Goncalo Ramos capitalised on a defensive error to place the ball past Zawieschitzky and score inside the final 20 minutes.
Milan were reduced to 10 men in the closing stages, after sub Filippo Terracciano was shown a red card for denying a clear goal-scoring opportunity, but the damage had already been done for the hosts and the Rossoneri eased to victory.
Salzburg’s defeat means that they have still never beaten AC Milan in their history, and Danny Rohl’s side will look to return to winning ways in the Europa League next week when they travel to Slovenian champions NK Celje.
Meanwhile, Ruben Amorim’s team will aim to secure back-to-back victories in Europe when they face AFC Bournemouth away from home.
Flashscore Man of the Match: Alphadjo Cisse (AC Milan)