The Italian side finished in a disappointing 13th position last season with coach Delio Rossi sacked for assaulting midfielder Adem Ljajic.
And Savic, who joined the Serie A outfit in a part exchange for Serbian defender Matija Nastasic over the summer, believes new manager Vincenzo Montella can turn things around.
"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."