United have won 11 FA Cups but last lifted the trophy in 2004 when they overcame Championship side Millwall 3-0.
Moyes said he is keen to end the drought starting with their Premier League opponents in the third round.
"To have not won the FA Cup for 10 years is something which is a bit of a surprise but it's something which we'll try and do," Moyes said.
"We're trying to win all the competitions we enter which we're sure most people are trying to do but as far as I'm concerned we'll do everything we possibly can.
"The cup is important to me, it always has been but I'm sure it's important to every club."
The Scot admitted however it has been a difficult period for the squad in terms of injuries, and that Ashley Young had hurt his shoulder in the previous match against Tottenham.
"The challenge which (Hugo Lloris) had on Young is going to keep Young out now for a time, so it's something we're going to have to look at," Moyes said
"We've got some other injuries as well that we'll just need to look at and see how they are over the next few days."
