The striker had already slashed an ambitious first-time volley wide before he opened his Milan account 10 minutes into the tie, Diego Laxalt's hopeful long punt putting his team-mate in the clear.
Piatek – who had scored four goals for Genoa in his maiden Coppa Italia appearance, against Lecce in August – kept his cool to convert, opening his body up to bend a right-footed effort beyond Meret.
He doubled both his tally and Milan's lead 17 minutes later, only this time his attempt with the same foot came when he was going away from goal, Piatek cleverly keeping the ball away from Kalidou Koulibaly as he cut inside from Milan's left before unleashing a drive that flashed into the net.
Napoli, in contrast, struggled to create opportunities despite dominating possession. Lorenzo Insigne missed an excellent chance to equalise for the visitors soon after Piatek had broken the deadlock, carelessly sending a first-time effort straight at Gianluigi Donnarumma
Alessio Romagnoli's block denied Arkadiusz Milik tapping home a loose ball following a corner after the break, with Milan holding firm to keep a clean sheet and become the first team through to the last four.
What does it mean? Milan continue to be up for the cup
Losing finalists in 2016 and 2018, Milan may hope this may be their year in the competition. Piatek's goals may be the added ingredient they need to lift the trophy for a sixth time, with boss Gennaro Gattuso a member of the team who last did so, back in 2003.
Piatek up and running for Milan
A reported fee of €35million was needed to bring in Piatek from Genoa, who made a healthy profit on a player they had only signed themselves in June. Still, the 23-year-old may just be the answer to Milan's scoring problems - they scored just 56 goals in Serie A last season - if he can build on a confidence-boosting brace.
Insigne off target as Napoli fall short
For a second successive season, Napoli's cup hopes came to an end at the quarter-final stage. Unlike the clinical Piatek, Insigne fluffed his lines in front of goal in the opening half, only locating Donnarumma after a cross had broken kindly into his path.
What's next?
Napoli will hope to bounce back from their Coppa exit when they host
Sampdoriain Serie A on Saturday, with Milan in action the day after, travelling to take on
Roma in a match with top-four ramifications.
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