Ancelotti was axed on Thursday after the comprehensive 3-0 UEFA Champions League loss to Paris Saint-Germain proved the final straw following an underwhelming start to the season.
The Italian guided Bayern to another Bundesliga title in his season in charge last term but rumours of player unrest and problems behind the scenes have derailed the German giants, who find themselves third in the league.
Ancelotti has already been linked with a return to Milan, where he won Serie A, Champions League, Coppa Italia, Supercoppa Italiana, UEFA Super Cup and FIFA Club World Cup titles.
However, Milan are happy with Montella, despite a topsy-turvy start to 2017-18 after a major and expensive overhaul of the squad during the off-season.
"It was sad, because Ancelotti proved himself to be one of the best coaches in the world and we don't know why this decision was made," Mirabelli told Sky Sport Italia.
"We are disappointed as fans of sport, but we have a coach who is younger than Ancelotti and just as talented. We believe in this project with Montella and want to reach the end of this journey with him.
"I understand these things are very tempting for you to write about, but everyone does their own job. We say that we are holding on tight to Montella."
Milan, who were stunned by Sampdoria in the Serie A last week, were less than convincing in a 3-2 win against Rijeka in the Europa League on Friday (AEST), having surrendered a two-goal lead.