The Nerazzurri finished ninth in Serie A last season, missing out on the automatic bye to the fifth round they had become accustomed to receiving and finding themselves hosting Serie B minnows Trapani at San Siro in early December.
The Sicilians, managed by lifelong Inter fan Roberto Boscaglia, did themselves no favours in the early exchanges as goalkeeper Ricardo Marcone's own goal gifted the 13-time cup winners the upper hand.
Ishak Belfodil then finished low past Marcone before Saphir Taider's penalty gave Inter what seemed like an unassailable 3-0 lead at the break.
But Cristian Caccetta earned cult hero status for Trapani with a shock goal as play resumed, the midfielder seizing on Antimo Iunco's pass before rifling beyond a bemused Juan Pablo Carrizo.
Giuseppe Madonia is another name Clarets fans will certainly remember fondly as his last-gasp goal from the bench put Trapani within touching distance of forcing extra time against the 2010 treble winners.
It was not to be for the plucky southerners, though, and Inter will travel to Udinese in the next round.
Chievo booked a last-16 trip to Fiorentina by brushing aside Serie B strugglers Reggina 4-1 at the Stadio Marc'Antonio Bentegodi.
Alberto Paloschi showed his top-flight class to score twice inside 25 minutes, his first goal arriving from the penalty spot, but nine minutes before the break Matteo Gentili halved the deficit from a corner.
When play resumed, Matteo Ardemagni re-established Chievo's two-goal lead with a deflected finish and Victor Da Silva's late header put the outcome beyond doubt.
Atalanta next visit Serie A high-flyers Napoli after edging out Sassuolo 2-0 in Bergamo.
The Neroverde managed to reach the break on level terms with their hosts but succumbed to second-half strikes from Moussa Kone and substitute Giuseppe Di Luca.