The Toffees certainly had the better of the first half as Tim Cahill hit the post, Paul Robinson saved superbly from Leon Osman and Kevin Kilbane fired just wide.
Noe Pamarot headed home what turned out to be the winner after 53 minutes, Osman then hit the post but that was about that from the home side.
Moyes said: "All credit to the lads, they worked really hard. They kept trying and worked terrifically well in the first half, couldn't take our chances and then we lost a goal that made us chase the game.
"We've done great, the players have done great so there is no shame in this one.
"Anybody who was here today will know that Everton deserved a win, I don't think there's any question about that, but we didn't."
Everton boss David Moyes was delighted with the effort of his side despite our 1-0 win at Goodison Park.