Antonio Conte's charges squeezed through 9-8 in the shoot-out after Sebastiano Esposito's second-half strike for the visitors had cancelled out Lorenzo Lucca's opener on 28 minutes.
The last-16 tie went straight to penalties after the 90 minutes, as per Italian Cup rules.
Cagliari's fourth penalty taker, Mattia Felici, was the first to blink, his attempt crashing against the crossbar.
But Brazilian David Neres missed his chance to be the match-winner as his next effort was easily saved by Elia Caprile.
Napoli goalkeeper Vanja Milinkovic-Savic made his first save of the shoot-out at the 10th time of asking as he denied Zito Luvumbo, before Alessandro Buongiorno netted to send the hosts through.
In the last eight, Napoli will face the winners of the tie between Como and Fiorentina, scheduled for the end of January.
Elsewhere, Atalanta ran out 4-0 victors at home to Genoa.
Inter Milan were equally handsome winners as they downed second-division Venezia 5-1 at the San Siro, with France international Marcus Thuram grabbing a brace for Cristian Chivu's side.
The round of 16 continues on Thursday as defending champions Bologna host Parma.