The Italians served up a 2-0 victory over the Republic of Ireland in Poznan on Monday just when they needed it most, with their hopes of reaching the quarter-finals in doubt.
Goals from Antonio Cassano before the break and substitute Mario Balotelli at the death were enough to claim the points and the win. And, coupled with Croatia's 1-0 defeat by Spain, the Italians secured second place in Group C.
Prandelli said: "I'm happy for the team and I'm happy for the squad. I am lucky enough to work with lads who are very serious and also lads who have won a lot of trophies, and they are still very motivated.
"My task in this is not that huge because they are the ones who determine the actual team spirit. The players have a lot of quality. First and foremost, they really need to show intensity and then their quality will start to show itself."
Italy certainly did show their quality on the night, but it was matched, for the first time in the tournament, by the Republic's battling spirit, for around 35 minutes at least.
It was then that Cassano managed to glance Andrea Pirlo's corner past goalkeeper Shay Given and although the Republic of Ireland rallied once again, they had midfielder Keith Andrews sent off for a second bookable offence just seconds before Balotelli made sure of the points with a last-minute volley.
Trapattoni proud of the Irish
Manager Giovanni Trapattoni said: "This evening, we played much better than in the games against Croatia and Spain. We played with more personality. The players respected what we said and played with creativity and commitment.
"We committed little mistakes on two corners, which has not been our habit, but I am proud of our commitment."
Republic of Ireland will head home tomorrow with their tournament over, but Trapattoni is tipping his native country for an extended stay in Poland and Ukraine. The 73-year-old Italian said: "Of course qualifying tonight, they have got even more confidence in them.
"Italy are well aware of their strengths and just how good they are, technically speaking. We are always talking about how many skilful and creative players they have. We talk about the Spanish players and people like [Lionel] Messi, but Italian players have a very good technique and very few passes go astray and of course, you grow in confidence when this happens.
"You can't rule them out from anything. They could go the whole way. There are other strong sides like Germany and England, but some strong sides will go home and Italy have the quality to go far."