The Uruguayan striker opened the scoring with a penalty he himself had earned before then setting up a clash with Siena in the last four with a second right at the death
Edinson Cavani struck twice for the Partenopei. The Uruguayan international fired home from the penalty spot after being felled by Thiago Motta before sealing the win in the dying seconds.
Inter went into the game on the back of an eight-game winning streak in all competitions but coach Claudio Ranieri made several changes to the side which defeated Lazio over the weekend and the visitors struggled during the opening 45 minutes to contain a full-strength Napoli lineup.
Indeed, Marek Hamsik had already had a sight of goal when he produced a wonderful crossfield ball in behind Christian Chivu. Christian Maggio controlled it brilliantly but then somehow contrived to allow Luca Castellazzi to deflect his subsequent shot over the bar.
Napoli, so renowned for its counterattacking prowess, continued to dominate possession and territory, though, with Inter threatening only sporadically. Indeed, its only effort on goal of note was a low strike from the ineffective and clearly frustrated Wesley Sneijder that was easily saved by Morgan De Sanctis.
Inter’s primary concern was keeping Napoli at bay and Chivu was not proving helpful in that regard.
The hapless Romanian was booked for hauling down Lavezzi after miscontrolling horribly in midfield before then clumsily colliding with Cavani just outside the area.
The Nerazzurri were looking vulnerable down the left-hand side, too, and Salvatore Aronica really should have opened the scoring when Hamsik picked him out with a fine cut-back but the defender shot straight at Castellazzi.
Inter showed greater ambition at the start of the second half and Thiago Motta went desperately close to getting on the end of a corner from the left-hand side.
However, that he failed to do so proved crucial because less than two minutes later, the Brazilian-born Italy international tripped Cavani in the area and the latter stepped up to convert the spot kick which followed to give the home side a deserved lead.
Inter was appealing for a penalty moments later when Diego Milito appeared to be dragged down by Maggio in the box but the officials saw no foul.
The visitors had secured a foothold in the game, though, and after Sneijder had fired just wide from distance, the Dutchman put Ricardo Alvarez straight though only to see the Argentine side-foot the ball straight at De Sanctis.
Inter bossed the final quarter of the game and could so easily have forced extra-time but De Sanctis frustrated the Nerazzurri with a wonderful double-save from Milito and then Esteban Cambiasso.
Napoli then sealed a semifinal showdown with Siena when Cavani finished in typically clinical fashion after a typically lethal counter from the Partenopei.