Luis Muriel stole the show by coming off the bench and firing home a long-range winner eight minutes from time, after Luis Vargas had cancelled out Antonio Di Natale's opener.
While Vincenzo Montella's men registered an away goal on their trip to the Stadio Friuli, Udinese will go into the second leg on Tuesday with the upper hand.
Coach Guidolin is hopeful his charges can reach the final of the competition for the first time since 1922, telling Rai Sport: "The dream continues, even if everything is still open and in the balance.
"I liked my team, but I also really liked Fiorentina. I think they, Juventus and Roma play the best football in Italy.
"We know it will be tough, but we're still cradling that dream. I don't know if we will get there, but we will certainly try."
Udinese's Coppa Italia campaign has been a bright point in an otherwise disappointing campaign for the club.
Despite sitting 15th in the Italian top flight, Guidolin is confident Udinese can achieve something this season so long as they remain committed to the cause.
"I am in love with this team and these players, so I am proud to lead them day by day towards a great objective," he added.
"While they are here, though, they must have the Udinese jersey glued to them, care for the club and fight for these colours. If they are to achieve something in future, then they will learn it here first."
