Burnley had been typically resilient in defence, while posing few questions of Everton, and it took a high-quality finish to separate the sides.
After Burnley winger Dwight McNeil was robbed by Gylfi Sigurdsson, Djibril Sidibe whipped in a cross from the right which Calvert-Lewin met with a superb angled diving header which flew in off the inside of the far post.
Sidibe had gone closest to breaking the deadlock with a low drive in the 32nd minute which Burnley keeper Nick Pope did well to keep out with his leg.
Italian Ancelotti, a Champions League winner with AC Milan and Real Madrid was appointed by Everton on Saturday to replace the sacked Marco Silva.
"Today, for me, was really special. The reception was fantastic here at Goodison Park. The atmosphere of the Premier League does not change. It was a special day, absolutely," said Ancelotti, who was sacked by Serie A club Napoli this month.
"We deserved to win. We were a little bit slow from the back in the first half - not so efficient on the counter attack. In the end, the performance was good. Not the top, but enough," he said.
Burnley boss Sean Dyche was left to reflect on 20-year-old winger McNeil's sloppy error.
"We never really looked in too much trouble and then we wanted to progress in the second half. It was a giveaway in the second half and they nicked it," he said.
"It was a poor mistake, Dwight was trying to play inside football and it was not on. He is a fantastic footballer and he will learn".