The Azzurri took the lead through Andrea Pirlo but Henrik Mkhitaryan levelled the scoring, though second-half goals from Daniele De Rossi and Pablo Osvaldo ensured three points for the Euro 2012 finalists.
The Italy boss admitted his team struggled at times against the unfancied opposition but deserve credit for fighting to a triumph on Armenian soil.
"We struggled to get things going at times, but overall I'm satisfied," Prandelli told reporters after the match.
"I can't say this enough: this is a tough group. But we played a great game and showed some character."
Prandelli's side face Denmark on Tuesday and sit top of Group B after three games, on seven points after wins over Malta and Armenia and a draw against Bulgaria.
The Italian coach also complained about the Armenian goal, adding: "When two players collide heads, you stop play. The referee didn't do it and that's why I protested."