Udinese progressed to the last four of the competition with wins over both Milan clubs and have claimed an advantage in their two-legged semi-final.
Despite Fiorentina dominating large periods of last Tuesday's first meeting at Stadio Friuli and creating more goal-scoring opportunities, the hosts completed a smash-and-grab win late on, aiding their bid to reach the Coppa Italia final for the first time since its opening year in 1922.
Antonio Di Natale's 36th-minute opener was cancelled out just before the break by Juan Manuel Vargas, whose goal gives Fiorentina a lifeline going into the second leg, but Luis Muriel netted Udinese's winner eight minutes from time.
Vincenzo Montella's men go into the tie on the back of a 2-0 home win over Atalanta, secured through goals from Josip Ilicic and Rafal Wolski.
Udinese coach Francesco Guidolin believes his side are approaching their best form following Saturday's 3-0 triumph over Chievo, but expects his men to continue improving.
Guidolin told Sky Italia: "We knew that we were having difficulties (against Chievo), but we remained calm.
"During the second half we increased our playing pace, acted more lively and pushed higher up the pitch. After the first goal, everything became easier. I give my compliments to Chievo, I just think our advantage became a little too heavy for them.
"I hope that the team continues like this, as now we're showing our true qualities. We must keep pushing and not let up even a little. We want to improve in this league that has now become more beautiful."
