Giampiero Pinzi stressed the importance of Udinese's 1-0 victory over Walter Mazzarri's men, especially considering they lost their opening game of 2014
Maicosuel's tap-in in the first half was enough to see the Bianconeri into the last eight, to face either AC Milan or Spezia Calcio, and consign the Nerazzurri to back-to-back losses.
As well as insisting the cup victory was for iconic skipper Di Natale, the tough-tackling midfielder praised the way Udinese fought for the win against the Serie A giants having been beaten 3-1 by Hellas Verona at the weekend.
"It's a great victory," the 32-year-old told Rai Sport after the Stadio Friuli game. "We came into it after a stinging defeat at home and we had to show what we could do against a great team like Inter.
"In the second half, we focused on our defence for our benefit and it helped produce a very important result. We prepared very well for the game and did well to shut down attacks and prevent Inter from breaking us down.
"Di Natale is a champion who has made us rejoice, taken us to Europe. We're dedicating this victory to him, he's our symbol and our captain and we want him to stay for many more years here."
Francesco Guidolin's men's triumph in the last-16 stage of the Coppa Italia sees them join Fiorentina, Juventus and Roma in the next round.