The Blues secured a point after coming back from two down to net in the closing stages through Tim Cahill and Leon Osman.
But the manager wanted all three points and was left to rue two goals conceded from set-pieces for leaving the Toffees facing an uphill struggle in the second period.
Moyes told evertonTV: "We didn't play very well but I don't think we deserved to be 2-0 down. Goals change games but I didn't think we played terribly badly - we just lacked a little devilment in the final third and our play was a little slow. But we got better in the second half.
"It was two set pieces today and two set pieces against Stoke that have caused us problems. Those things can be corrected but we've been making a few individual mistakes as well and if people keep making mistakes they have to be accountable because we can't continue conceding goals the way we're doing just now.
"In the second half we got better and didn't drop off after the Uefa Cup. The players worked really hard but we know that is how they are. But I'm disappointed we didn't win the game."