Inter – who have not won a trophy since claiming the Coppa Italia in 2011 – entered the competition in the third round after last year's disappointing Serie A finish.
With the top eight sides last season's Serie A not coming into the competition until the fifth round, Inter's placing of ninth saw them at the risk of a shock loss in the early stages.
Walter Mazzarri's side have lost just once this season – that being to the second-placed Roma – and sit fourth in the Serie A table.
They were not at their best on Sunday though, drawing 1-1 with Sampdoria at San Siro in a performance that Mazzarri claimed was their poorest of the season.
Inter's injury list is lengthy at the moment, with Diego Milito, Mauro Icardi, Cristian Chivu, Yuto Nagatomo and Walter Samuel all sidelined.
Trapani – who were champions of the third tier last season – have taken time to find form in the current campaign but will be buoyed by three successive victories.
The latest of those came on Saturday as Simone Basso's 68th-minute strike gave them a 1-0 triumph over Virtus Lanciano.
