Two goals in the space of five minutes from Riccardo Montolivo and Daniele De Rossi had Gli Azzurri 2-0 up, although Denmark got a goal back in first-half stoppage time through William Kvist Jorgensen. Then, just 18 seconds into the second half, Osvaldo was shown red for violent conduct after an incident involving Denmark's Nicolai Stokholm.
However, Mario Balotelli, making his first appearance for Italy since UEFA EURO 2012, fired in a third in the 54th minute to put the result beyond doubt. The win saw Italy move four points clear at the top of Group B, and Prandelli could find few faults with his team's performance.
"We had some initial difficulties, but then we began to play some good football and we deserved the two goals," he told the Italian Football Federation's website. "We used up a lot of energy playing a man down, but I saw a great spirit in the ten still out there and we fought with determination and order.
"And for this, I have to congratulate the lads. We are now a bit more relaxed about our position in the group but we have to improve a lot."
It was a happy return to the San Siro for former Inter Milan striker Balotelli.
"I was hoping to win this game," he said. "We took the lead but things then became a little complicated when they scored at the end of the first half. My goal was instinct. I just tried to anticipate the goalkeeper. It's always special to play at the San Siro. It's a great stadium and gives you great emotions."