The defending cup holders were not unduly troubled by the Rossoneri - who were reduced to 10 men - after they sealed victory in the first half
Milan have proved tricky opposition for Juve this season and followed up the 1-0 league win at San Siro in October with a penalty shoot-out triumph in the Supercoppa in December.
But holders Juve burst out of the blocks in Wednesday's contest in Turin, their excellent first-half showing proving enough to set up an enticing semi-final with Napoli despite a spirited showing from a Milan side forced to play the final 36 minutes a man down.
Paulo Dybala gave Juve a 10th-minute lead with a close-range volley before Miralem Pjanic made it 2-0 with a fine free-kick 11 minutes later.
After a dreadful first half, Milan found their stride and pulled one back through Carlos Bacca's stunning 53rd-minute effort.
However, Manuel Locatelli was then sent off for a second bookable offence and, though both teams had chances in a hugely entertaining second half, Juve stood firm to keep their bid for a third straight league and cup double alive.
Neither goalkeeper had really been threatened before Dybala struck, Gianluigi Donnarumma helpless to prevent him from firing Juve ahead with his eighth goal of the season.
Kwadwo Asamoah's left-wing delivery was cleverly flicked into Dybala's path and he lashed a fierce volley into the bottom-right corner.
Milan continued to be posed problems by Asamoah and the second goal came as a result of more good work from the Ghanaian.
Asamoah was clipped on the edge of the area by Juraj Kucka and Pjanic took full advantage from the resulting free-kick, curling beyond the outstretched Donnarumma in superb fashion.
Juve looked to have a third when Sami Khedira found the net on the rebound after Gonzalo Higuain had latched onto a wonderful cross-field pass from Asamoah and seen his shot saved by Donnarumma, but the assistant's flag denied the Germany midfielder.
Milan finished the first half on the front foot and halved the deficit eight minutes after the restart.
Khedira robbed Andrea Bertolacci following Kucka's knock down, but his clearance only went as far as Bacca, who met the ball with a magnificent hooked volley that left Neto with no chance.
But Milan's task of completing the comeback was made more difficult moments later when Locatelli was shown a second yellow for a rash sliding challenge on Dybala.
Mario Mandzukic then twice wasted excellent chances to restore Juve's two-goal advantage, heading narrowly wide on both occasions.
Milan still posed a threat despite their numerical disadvantage and Neto had to parry strongly to prevent Kucka from levelling matters with a driven free-kick.
Gerard Deulofeu was brought on for his debut 10 minutes from time and, after Donnarumma had done well to keep out a Pjanic volley, the on-loan Everton man spurned a great chance to level when he got in behind the Juve defence only to produce a tame effort that Neto easily saved.
Milan were unable to threaten any further and, while Juve's domestic domination continues at pace, Vincenzo Montella's men will have to focus fully on their quest to bridge the seven-point gap to third-placed Napoli in Serie A and Champions League qualification.