The Toffees take on Fulham at home tonight in the fourth round of the competition.
"The FA Cup is our last realistic chance of winning a trophy this season, and it's down to me and the players to give the supporters something to get excited about," Moyes said.
"We will have to play a lot better than we did against Blackburn.
"Playing better will lift the mood – winning football matches is always the answer and I take full responsibility for that."
Everton are currently 14th in the Premier League with 26 points from 22 games.