The two sides played out a 2-2 draw after extra time at the Stadio Diego Armando Maradona before Cremonese, without a win in Serie A all season, prevailed 5-4 on penalties.
The visitors took the lead through Charles Pickel in the 18th minute but Juan Jesus and Giovanni Simeone regained control of the tie for the hosts before substitute Felix Afena-Gyan sent the game to extra time with a powerful header three minutes from time.
Stanislav Lobotka proved to be the fall guy in the shootout as he missed his kick before Afena-Gyan struck the decisive penalty to take Cremonese through.
Napoli are chasing a first Serie A title since 1990 and are also through to the knockout rounds of the Champions League but the Coppa Italia represented another chance of landing silverware in what could prove to be a memorable season in the club's history but manager Spalletti was left with a bitter taste in his mouth.
"We are sorry," coach Spalletti told Mediaset.
"There is so much disappointment.
"We always have to go through the management of the game, of the ball, and we have not been as good as we usually are.
"But then the game was in complete safety because they, apart from a few balls to the attackers, could not do much. We conceded goals in a situation where we lost two balls that were going out.
"Then we scored two goals and wasted several chances, but unfortunately that's how it works. When you question a game that was under management then it becomes complicated."
