Foy opted to red card City captain Vincent Kompany with just 12 minutes on the clock for a challenge on United winger Nani.
Mancini was visibly bemused by the decision at the time, but has instead chosen to focus on his team's performance after the break.
"It's better we talk the game, not the referee. Fantastic performance, played well in second-half," the Italian told ITV Sport.
Asked whether his side may have got a positive result, Mancini replied: "A different game, clearly."
City won the FA Cup last season with a 1-0 victory over Stoke City in the final.