The 21-year-old, a €38million signing from Porto, and Riccardo Montolivo each scored twice, while second-half finishes from Fabio Borini and Luca Antonelli put Vincenzo Montella's side on the brink of the group stage.
Milan, who handed a competitive debut to new captain Leonardo Bonucci, were in control from the outset and scored all three of their goals in a rampant first 27 minutes.
Silva converted two clinical finishes and Montolivo scored a tap-in to end a three-year wait for a Rossoneri goal and send the home crowd, who numbered in the region of 40,000, into raptures.
Borini buried his first goal for the club before Antonelli made the most of more slack goalkeeping and Montolivo fired in a fine sixth to add gloss and all but assure Milan of a return to European competition proper for the first time since the 2013-14 season.
Antonelli and Franck Kessie came close before Silva broke the deadlock with 13 minutes played, the Portugal international reacting quickest to a loose ball in the box to drill home his first goal at San Siro.
Montolivo made it 2-0 with 25 minutes played. Kostadin Zahov failed to parry Suso's free-kick far from his left-hand post, and the Italy veteran pounced to tap home his first goal since April 13, 2014.
Milan were in cruise control by this point and Silva got his second within two minutes – and in some style. The striker showed good close control to turn with the ball inside the box and battled his way through two challenges before placing a fine finish into the bottom-right corner.
Silva thought he had completed his hat-trick three minutes later as he turned Antonelli's flick-on towards goal with his thigh, but the ball struck the base of the post before Zahov just clawed it away before it crossed the line.
But with 67 minutes played, Borini got his first goal for the club. A fine team move involving Montolivo, Suso, Andrea Conti and Silva ended with the latter scuffing a first-time shot into the path of Borini, who converted from 10 yards out.
Just a minute later, Milan had their fifth, with Antonelli arriving to tap the ball in at the far post after Zahov parried Suso's low shot from the right straight into the full-back's path.
Zahov partly made amends with a brilliant reaction stop to prevent Silva completing his hat-trick with a backheel but he was powerless to stop Montolivo grabbing his second with five minutes left, the midfielder collecting Ignazio Abate's pass on the turn before blasting low into the net past the keeper's outstretched leg.