The Rossoneri shot out of the blocks in Vienna as Calhanoglu netted the opener before twice laying on first-half goals for Silva, who completed his treble after the break, with substitute Suso adding a fifth.
Vincenzo Montella's men had made a bright start to the season, with six straight wins including Europa League qualifiers, but that run was halted in spectacular fashion with a 4-1 Serie A loss at Lazio last weekend.
However, returning to the full draw of European competition after three years away, Milan hit their stride again here with a dominant display.
Wien simply had no answer for Calhanoglu or the movement of forward pair Silva and Nikola Kalinic - making his full Rossoneri debut - in a one-sided opening 45 minutes, but they did muster a response shortly after the restart with Alexander Borkovic's header.
Silva's third - his fifth of the campaign - followed, however, and Suso arrived from the bench to instantly add his own name to the scoresheet, capping a fine win.
Only a smart low Osman Hadzikic stop kept Calhanoglu from opening the scoring with Milan's first attack after he had ghosted onto the end of Ignazio Abate's cross.
But the Turkey international did not have to wait long to break the deadlock, leading a counter and trading passes with Kalinic before blasting a strike across Hadzikic into the top-right corner.
And Calhanoglu was quickly involved again to tee up the second. Refusing to give up on Franck Kessie's pass, the former Bayer Leverkusen man scooped the ball away from Abdul Mohammed to slip in Silva for a cool finish.
Gianluigi Donnarumma was almost caught out by Christoph Monschein's touch from a Raphael Holzhauser free-kick, but yet another Calhanoglu-inspired break soon led to the third goal before the 20-minute mark.
Another wonderful pass picked out Silva on the run, with Kalinic this time peeling away to create a distraction, and Hadzikic was easily beaten again.
Wien kept the visitors out until the interval, though, and then, after testing Donnarumma through Lee Jin-Hyun, hit back with a stooping Borkovic header.
Kalinic and Calhanoglu each then fired efforts off target, but it was Silva, receiving a cute Kessie pass, who restored the three-goal lead and earned the matchball with Milan's first European hat-trick in almost 11 years.
Suso - Kalinic's replacement - had been on the pitch for just over a minute when his long-range blast deflected off Mohammed and over Hadzikic for the fifth and final goal.
Key Opta stats:
- The last time AC Milan scored 5+ goals away from home in European competitions was in October 1993, against Copenhagen.
- The Rossoneri have scored 15 goals over the five Europa League matches they have played this season, conceding only one in the process.
- Hakan Calhanoglu has both scored and delivered an assist for the third time in European competitions: the previous two were for Bayer Leverkusen in 2015 (vs BATE Borisov) and 2015 (vs CSKA Moscow).
- Suso netted his first goal in Europa League, he already featured in this competition, with Liverpool back in 2012-2013.