Ezequiel Lavezzi blasted over the only missed spot-kick as Inter prevailed 5-4 on penalties and kept alive their dream of a second straight treble.
Hosts Napoli had much the better chances in the 120 minutes with Marek Hamsik going closest when his shot was saved and his subsequent header was cleared off the line.
Inter battled hard without much penetration and lost Dejan Stankovic and Ivan Cordoba to injury.
Two goals by striker Alexandre Pato earlier downed Sampdoria 2-1 to seal Milan's place in the two-legged semi-final stage, where they will meet Palermo after the Sicilians overcame Parma on penalties on Tuesday.
The Brazilian, left out of the Serie A leaders' win over Cesena on Sunday, struck the first after 17 minutes and converted Dutch debutant Urby Emanuelson's assist five minutes later to make it 2-0.
Stefano Guberti headed the Sampdoria back into the game after the break but Milan, who also gave new signing Mark van Bommel a first start following the uncompromising Dutchman's move from Bayern Munich, hung on.
With 12 minutes left Milan brought on former Samp forward Antonio Cassano, who left the Genoa-based side in January in acrimonious circumstances after being dropped for swearing at the club president.
Juventus host AS Roma on Thursday in the last quarter-final.
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