Rodwell was a second-half substitute in the Wembley defeat.
He said: "We knew it was going to be a tough game, especially on such a big pitch but we just weren't at the races today.
"We're going to have to bounce back now.
"We have a game on Tuesday and we want to make sure we finish in second place."
Asked if the speculation surrounding Mancini's future affected his side, former Everton midfielder Rodwell added: "No. I don't know anything about it."