The Azzurri moved three points clear at the top of Group J after leaving the hosts reeling with three goals in the space of 10 minutes.
Nicolo Barella, Lorenzo Insigne and Leonardo Bonucci were all on target as Italy raced into a three-goal lead after just over half an hour.
But they should have added to their tally, with Insigne and Federico Chiesa guilty of missing chances after the break.
“We were good to address the game in the way we wanted from the start,” Mancini is quoted as saying by UEFA.
“We were good at defending high, but we were less good in the second half when we should have added the fourth.”
Nevertheless, Insigne is excited to be part of the current Italian side.
He said: “The coach is creating a great group. We need it if we want to go far.
“We all love to play for this national team and I think it’s visible on the pitch.
“There are several youngsters and some veteran players with experience. It’s really the perfect mix.”
Fellow striker Andrea Belotti, who set up the first goal for Barella, added: “I am happy with my performance. It is difficult to find the right words because for me playing in the national team is always a dream.
“Mancini was a bit angry because in the second half we did not press as we did in the first and we saw the difference.
“We always have to improve. When you lower the pace, football is strange; even today we risked conceding. The important thing is the victory, but we have to work to limit our mistakes.”
Defeat saw Greece slip behind Finland, who now occupy second spot in the group.
Provided by Press Association Sports
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