The Italian side finished in a disappointing 13th position last season with coach Delio Rossi sacked for assaulting midfielder Adem Ljajic.
Montella was appointed with the goal of transforming Fiorentina's fortunes and Savic is eager to be a part of his coach's plans, after joining from Manchester City as part of an exchange deal for Matija Nastasic.
"I made my debut on Sunday and I'm glad it coincided with a win for my team," he told Fifa.com
"After the bureaucratic problems I finally played, and this is the beginning of a beautiful adventure.
''I am pleased with my form and a debut victory. I am very happy at Fiorentina.
"I hope that this season there may be room for me and the team will be able to reach the top five, so that we can bring Fiorentina into Europe."