Fiorentina were beaten 2-0 in the second leg of their semi-final against Sevilla on home soil following a 3-0 loss in the first leg.
Montella's men were subjected to jeers from their home fans as they slipped to yet another heavy defeat against the La Liga club.
"We certainly weren't fortunate over the two legs, as Sevilla scored almost 100 percent of their chances and we got 0 percent," Montella told Sport Mediaset. "I thought Fiorentina tried to play and score right to the final whistle, even though qualification was clearly out of reach, so they deserve more respect than they received.
"Considering the efforts, the results so far all season, I don't think this side deserved jeers. I think my team deserves more respect than they are getting at this moment from the crowd. They do not deserve to be derided by their own fans.
"Perhaps this proved the Fiorentina fans in the stadium tonight do not understand their dimension in the football world. It's a shame."
Montella also admitted that his future as manager of Fiorentina, who currently sit in fifth place on the Serie A standings, remains uncertain.
While fifth place will see them qualify for next season's Europa League, Montella made it clear that the Italian club will do everything in their power to try and earn a spot in the Champions League.
"I have a contract, we'll evaluate everything at the end of the season," he said. "The club may well even decide not to confirm me on the bench, if they are not satisfied. The club has known my thoughts for three months now.
"With the resources at our disposal, I think we achieved slightly more than we were really capable of this season.
"I hope to continue with players and indeed men of this character, plus for Fiorentina to have them in future."