Captain Matt Mills struck the only goal at Goodison Park to send the Royals into the quarterfinals.
Moyes, whose frustration was exacerbated by a season-ending injury to Marouane Fellaini, said: "We let the supporters down tonight.
"We didn't play well enough and we got what we deserved.
"After coming through the Chelsea tie and getting a home draw was all you could ever ask for.
"We got that but we never came up to scratch, apart from the first 10-15 minutes when we passed the ball well.
"We didn't have enough quality, enough craft or guile to break them down at the top end of the pitch."