Second-half goals from Aleksandar Kolarov and Sergio Aguero were not enough for the holders after captain Vincent Kompany had been sent off early on.
"I think this is important for us in the league. I think now we can win the title," Mancini told reporters.
"United have the top squad in England and if we can play like we did against them with ten players, with that same attitude and same strength in every game, we will win.
"I am disappointed to lose the game but I think we still took two steps forward after this because we have shown character."
City host Liverpool in the first leg of the Carling Cup semi-final on Wednesday.