Napoli's Stanislav Lobotka was the only player to miss a penalty, as Cremonese, who have yet to win in Serie A this season and are bottom of the table, went through to the next round.
Cremonese shocked the home supporters when they took the lead through Charles Pickel after 18 minutes, but two goals in three minutes from Juan Jesus and Giovanni Simeone before halftime calmed Napoli's nerves.
While Napoli controlled much of the second half, Cremonese equalised in the 87th minute, with Felix Afena-Gyan sending a header into the top corner.
Cremonese's Leonardo Sernicola was sent off for a second booking in extra time, but Napoli could not take advantage of the extra player and the game went to penalties.
Cremonese will face Jose Mourinho's AS Roma in the quarter-final on Feb. 1.