Napoli had so many opportunities to win it, but Ezequiel Lavezzi's shoot-out miss sent Inter through to the Coppa Italia semi-final.
Both sides are taking this tournament very seriously and fielded full-strength line-ups for the quarter-final.
Napoli thought they had taken the lead after 12 minutes when Douglas Maicon lost the ball in midfield, sparking a counter-attack. Ezequiel Lavezzi threaded through for Andrea Dossena, whose cross found Edinson Cavani to tap in at the back post.
The referee disallowed the goal for an offside position, but even the replays do not entirely clarify the extremely tight decision.
Dejan Stankovic had a great chance, flashing his volley across the face of goal from a tight angle, but a Cristian Chivu howler in the penalty area gave Marek Hamsik a chance to have his shot deflected over the bar.
Morgan De Sanctis rushed into midfield for a diving header clearance to stop Goran Pandev, while Javier Zanetti came to Andrea Ranocchia's rescue after Cavani skipped past the Italian.
Chivu really should've scored on 37 minutes when a corner found him for a totally free header from six yards, but he mistimed the jump and ended up connecting with his shoulder instead.
From the corner, Cavani went on a blazing counter-attack and sent Lavezzi clear, but El Pocho hit a weak effort at Luca Castellazzi rather than pass across to the unmarked Uruguayan.
Inter had another incredible scoring opportunity moments later, as it was end-to-end stuff. This time Maicon's cross found Esteban Cambiasso for the volley from seven yards, but De Sanctis somehow got his fingertips to turn it on to the crossbar!
After the break Lavezzi knocked down a long ball and rolled across for Hamsik, but the Slovak's first touch let him down in the box.
Bad news for the visitors, as Stankovic collapsed with a suspected thigh strain. Christian Maggio felt he had been shoved in the back by Ivan Cordoba on the very edge of the box as he ran on to a Cavani assist, but the referee waved play on. It could've potentially been a red card for ending a clear scoring opportunity.
It was remarkable this game had no goals, as Paolo Cannavaro was unmarked at the back post and opted to chest down the free kick for Maggio rather than go for the header.
There was chaos on 83 minutes in the Inter box! First Hamsik's strike was beaten away by Castellazzi's outstretched foot from point-blank range, then a Ranocchia goalline clearance denied Hamsik's headed follow-up.
Michele Pazienza's header from a corner skimmed the upright, but neither side could find a way through and this quarter-final went to extra time.
There was further controversy when Lucio charged down a Hugo Campagnaro strike with his upper arm in the penalty area, as replays suggest he did move towards the ball.
Still nobody could break through. Lavezzi burst into the box following a defensive error and tried a dummy to send Castellazzi down, but the goalkeeper still managed to deflect the angled drive wide at the near post from point-blank range.
More penalty appeals for Thiago Motta's shirt tug on Cavani, but the referee thought it was six of one and a half-dozen of the other.
A free kick touched on for Cavani was charged down, as Napoli dominated extra time, but at the other end Pandev blasted over after good work from Zanetti.
Juan Zuniga thought he'd scored an unbelievable winner at the 117th minute with his scissor-kick volley at the back post, but Zanetti got there in time to charge it down! From the corner, Castellazzi needed two touches to get Cavani's header under control.
Ranocchia got his head to a Maicon free kick, but turned it well over, then Eto'o blasted off target from distance with the last kick of the game.
It needed a shoot-out to decide who would reach the semi-final. Cavani was first up and sent Castellazzi the wrong way.
Samuel Eto'o, Hamsik and Cambiasso converted successfully, but Lavezzi fired over the bar.
Pandev scored despite De Sanctis getting the right angle then Castellazzi thought he had saved from Zuniga, but the ball squirmed under his stomach to roll over the line.
Thiago Motta scored, so Hasan Yebda had to score to keep Napoli alive, but he didn't let the pressure get to him with a cheeky chip.
Chivu was last up for Inter and maintained their 100 per cent record. The holders reach the Coppa Italia semi-final for the eighth consecutive year.
Penalties: Cavani (N, goal), Eto'o (I, goal), Hamsik (N, goal), Cambiasso (I, goal), Lavezzi (N, missed), Pandev (I, goal), Zuniga (N, goal), Thiago Motta (I, goal), Yebda (N, goal), Chivu (I, goal)
Napoli 0-0 Inter (aet, 4-5 on pens)
Napoli: De Sanctis; Campagnaro, Cannavaro, Aronica; Maggio, Pazienza (Yebda 97), Gargano, Dossena (Zuniga 70); Hamsik, Lavezzi; Cavani
Inter: Castellazzi; Maicon, Cordoba (Lucio 83), Ranocchia, Chivu; Zanetti, Cambiasso, Thiago Motta; Stankovic (Mariga 62); Eto'o, Pandev