The Azzurri piled on three second-half goals to secure the win in the opening game of the Euros.
Juventus defender Merih Demiral netted an own goal to break the deadlock from Domenico Berardi’s cutback in the 53rd minute.
Italy added a second from Ciro Immobile on 66 minutes, with Lorenzo Insigne curling in a third in the 79th minute.
“I was hoping for a start like this, we are pleased to have played well and entertained the fans,” Mancini said.
“I hope the Italians had a good evening. We played well, we never gave up. In short it was an excellent match.
“We are aware that we are a good team. We beat Turkey, who are definitely not pushovers.
“We know we can still improve, even with young players who don’t yet have European Championship experience.”
In Group A, Wales take on Switzerland on Saturday, with the Italians already setting the pace.
“It was important to get off to a good start,” Mancini added.
“We produced an excellent performance against a team that wasnt as weak as they seemed.”