City won 2-0 at Eastlands against The Potters on Boxing Day in what was Mancini's first game since becoming City's new boss.
"I am very happy," Mancini told Sky Sports.
"I'm very happy with this match because we won a hard match.
"I said that the players today were fantastic and in a very tough situation we always worked.
"I think we must improve, but today I'm very happy for the players."