Moyes said: "Tomorrow is a night we need the fans right behind the team and supporting us the way they did at Chelsea.
"That was probably the best we've been able to give them this season, but you always hope the best is still to come.
"Let's hope it is in this game and future games as well.
"But it's the FA Cup - you never know what is going to happen. Throughout the years there have always been results you don't expect.
"We have to make sure this is one where we do expect to get Everton through.
"But I think Brian McDermott has done a great job down there with his team.
"He has steadied Reading and it wasn't too long ago they were a Premier League team, so we have got to be mindful of that."