The Viola must overturn a 2-1 first-leg deficit next week in Florence if they are to reach the Coppa final for the first time since 2001.
"We could have done much more," Montella told Gazzetta dello Sport. "We gave Udinese too much space and they punished us at the end."
The game looked to be heading for a 1-1 draw before Udinese substitute Luis Muriel struck the winner eight minutes from time.
"It was one of those games open to any result," Montella said. "We could have won but we ended up losing.
"We knew that Udinese would make life difficult for us because they are a great side.
"It is annoying to lose but we have to congratulate Muriel, who scored a very good goal.
"This result just means that we will have to be even more determined in the return leg."
Udinese coach Francesco Guidolin believes his team is one step closer to reaching the club's first Coppa final since 1922.
"Our dream continues," Guidolin said. "We know that there is still 90 minutes left to play and anything can happen.
"We know it's going to be tough to play in Florence but we are holding onto our dream.
"I don't know if we will achieve it but we will certainly try."
The return leg is at the Artemio Franchi stadium on Tuesday with the winner taking on either Napoli or Roma in the Coppa final on May 3.