Coppa Italia Wrap: Inter survive scare to advance
It took only four minutes for Inter to take the lead in front of their home fans, with Colombian international Freddy Guarin sneaking a free-kick in at the near post.
After dominating possession through the first half, Inter were rewarded in the 40th minute with a superb backheeled volley from Ishak Belfodil.
When Saphir Taider converted a penalty on the stroke of half-time, Inter took a comfortable 3-0 lead into the break.
But Trapani started their fightback on 55 minutes as Cristian Caccetta made a probing run to receive the long throw and finished through the legs of the goalkeeper from a tight angle.
Some poor Inter defending allowed Trapani's Giuseppe Madonia to volley home in the 90th minute but the underdogs could not force an equaliser in the dying moments.
In other matches, Alberto Paloschi's brace powered Verona to a comfortable 4-1 win over Reggina.
The 23-year-old striker scored twice in the first half as his performance laid the foundations for victory.
Elsewhere, two second-half goals helped Atalanta to a 2-0 home win over Sassuolo.