The game began promisingly for Napoli, when Lorenzo Lucca rose to head in a Matteo Politano cross in the 28th minute.
But the advantage did not last, in the 67th minute substitute Sebastiano Esposito capitalised on a loose ball, resulting from a rebound off a tackle, to level for Cagliari.
Under the rules of the Coppa Italia, the game went straight to penalties rather than extra time. The shoot-out proved gruelling: both sides converted their first nine kicks, before Cagliari's fourth taker, Mattia Felici, struck the crossbar.
Later in the sudden-death phase, Napoli goalkeeper Vanja Milinkovic‑Savic finally made a decisive contribution, saving Cagliari's Zito Luvumbo, before defender Alessandro Buongiorno slotted the winning penalty.
Napoli head coach Antonio Conte described the shoot-out as "interminable", but insisted "the team that deserved it the most went through."
Having survived the scare, Napoli now await the winner of the tie between Como and Fiorentina, with their quarter-final fixture likely in late January.
Meanwhile, in other Coppa Italia action, Atalanta thrashed Genoa 4-0 to reach the last eight while Inter Milan crushed Venezia 5-1.
AC Milan's resurgence under Massimiliano Allegri has prompted growing belief that Serie A may be entering a new "Super Max" phase, with the Rossoneri unbeaten against the league's biggest sides heading into December.
AC Milan go into tonight's clash against Lazio at San Siro knowing that a win will lift them to the top of Serie A, at least overnight, and once again the focus falls on Luka Modric.
Antonio Conte has insisted that Scott McTominay is not being sacrificed after the Scotland midfielder battled through a heavy knock in Napoli's hard-fought 1–0 win away to Roma.
Diego Simeone said Atletico Madrid "had a plan" to break Inter down and insisted his side executed it "to perfection" as they claimed a dramatic 2-1 win in the Champions League.
