The UEFA EURO 2012 runners-up may top Group B courtesy of two wins and a draw, but concerns over a lack of ruthlessness in the Azzurri attack remain, with a resolute Denmark side visiting Milan on Tuesday night.
Hrayr Mkoyan's handball allowed Andrea Pirlo to nudge Italy ahead from the spot in Yerevan, only for Shakhtar Donetsk's Henrik Mkhitaryan to drive in an equaliser in the 27th minute. But Pirlo turned provider in the second half, setting up Daniele De Rossi to head home, with Roma striker Pablo Osvaldo making the points safe ten minutes from time with his own header.
After the match, Prandelli told Figc.com: "As I've said before, our group is a difficult one and tonight proved that. We played a good game, showing a lot of character and taking a few risks.
"They came at us a few times, but we always had control of the game and were clinical in front of goal. We started well and created four great chances on goal early on. The team showed the will to win right until the very last minute, and that kind of attitude is very important."
He said: "It was an important goal that allowed us to relax a bit towards the end. I always go out there to give my best and, when the call comes, I give my all for both Roma and the national side.
"Hard work pays off and you could see that out on the pitch. You can say anything you like about me, but never that I'm not committed."
Zenit defender Domenico Criscito was delighted to return to the international set-up after a lengthy absence.
"It's difficult to come back after six months away, but wearing the blue shirt again was incredible," he said. "We played a great game, we started very well and we almost always had possession of the ball."