The Viola headed into the game in Florence trailing 2-1 from the first meeting, but prevailed 2-0 - thanks to goals from Manuel Pasqual and Juan Cuadrado - to secure a 3-2 aggregate triumph.
"Udinese were worthy rivals but I think over the two legs were were better and we deserved to go through," Montella said to Gazzetta dello Sport. "We are growing a lot and I saw a great team spirit.
"Our hunger was what made the difference. My players did a good job.
"We kept a clean sheet and gave nothing to Udinese. We are very happy for our fans and for our city.
"We have improved a lot throughout the season and we have taken a leap in quality."
Fiorentina will take on either Roma or Napoli in the final on May 3.
"They are both very strong teams so it doesn't matter who goes through because we expect a very difficult game," Montella said.
"The important thing is that we have reached the final, which was one of our aims at the start of the season.
"The fact that we have made it so far also guarantees us at least Europa League football next season."
Fiorentina are fourth in Serie A, three points behind Napoli, who are third and hold the last Champions League qualifying spot.
Roma are second in the standings, four points clear of Napoli, and have a game in hand.
Udinese coach Francesco Guidolin was disappointed with the outcome but felt his side left the competition with their heads held high.
The Friuli outfit were looking to reach the Coppa final for the first time since 1922.
"I am proud of my players," Guidolin said. "It wasn't easy to play in this stadium and against Fiorentina, who in my opinion are one of the best teams in Italy.
"You can risk being humiliated by Fiorentina but I thought we played at the same level as them.
"We played well and the only thing we were missing was to score.
"We leave the competition with our heads held high and wish Fiorentina the best of luck in the final."
Tuesday's outcome ended Udinese's hopes of playing in Europe next season.
Guidolin's side have struggled in Serie A so far and are unlikely to finish in the top five.
They lie 14th in the standings - 10 points adrift of Inter Milan, who are fifth and hold the last Europa League qualifying berth.
"We cannot perform miracles every year," Guidolin said. "There are teams in the league that are now doing what we used to, like Hellas Verona.
"It's a difficult season and we have a very young squad and must focus on Serie A."