Ancelotti is the first Everton boss to win the award since David Moyes in 2013.
The Italian claimed the accolade for the fifth time, having won it on four occasions while in charge of Chelsea between 2009 and 2011.
Ancelotti has guided Everton to the top of the Premier League after winning their opening four league matches for the first time since Harry Catterick’s team in 1969-70.
Calvert-Lewin’s goals and the fine form of Ancelotti’s new signings James Rodriguez, Allan and Abdoulaye Doucoure have revitalised Everton.
Praising his players’ for their impressive efforts, Ancelotti told Everton’s website: “I am really pleased to win this award. I think it is a great achievement for this month. I have to share with my team because they started the season really well.
“We hope to continue and to win this award again. I think the key point was the first game against Tottenham.
“Tottenham is a really good team, but we did well and won away. That victory increased our confidence and belief for this season. It was a really good start.”
Calvert-Lewin is Everton’s first Player of the Month since Romelu Lukaku in 2017.
The 23-year-old has scored six goals in four league games this season and netted on his England debut in the friendly win against Wales on Thursday.
“Dominic had a fantastic month. He deserved to be the player of the month. He is young but he has had a lot of improvement,” Ancelotti said.
“This last month, he has been really focused in the box and this is the reason he scored a lot of goals.”