Manuel Locatelli’s double and Ciro Immobile’s late strike sealed a dominant 3-0 win over Switzerland in Rome.
They returned to the top of Group A, after Wales’ 2-0 win over Turkey, following a 10th straight win without conceding.
Mancini said: “In the European Championship there’s France, Portugal and Belgium – one of those are world champions, another are European champions and the other are the number one ranked side in the world.
“These are teams who have been built over a number of years and it’s only natural they are further along than us but everything can happen in football, you shouldn’t take anything for granted.
“Every match is difficult, you always have to go out there and play. I’ve been fortunate in that I have had some very good players who like playing the game, they enjoy being out there and like taking risks.
“The players are the ones who deserve the credit. I try to explain my thought process and there is still a long way to go.”
Giorgio Chiellini who was later forced off injured, had an early goal disallowed by VAR for handball before Locatelli struck after 26 minutes when he started and finished a swift Italian move.
His second came seven minutes after the break when the Sassuolo midfielder collected Nicolo Barella’s pass and his fierce 20-yard shot found the bottom corner.
Italy were never seriously troubled by Switzerland, who face Turkey next needing to win to stand a chance of progressing, and Immobile sealed victory as Yann Sommer could not keep out his low drive from 25 yards with a minute left.
Switzerland boss Vladimir Petkovic said: “We played against a strong team, they caused us a lot of problems and we weren’t 100 per cent.
“We were insecure on the pitch which is normally not in our character. It’s disappointing, I have already talked to the team and from tomorrow we have to look ahead, be positive and ready for our decisive match against Turkey to have the chance to make the last 16.
“Italy are a great side and if they keep going as they have been doing they have everything it takes to make it to the semi-finals. There are other sides who are very good.”