The Toffees failed to hold onto a two-goal lead against Birmingham.
Moyes felt his side deserved a win over Birmingham, following their display in the first half.
“But for an inspired goal-keeping performance (from Joe Hart) and an incredible sloppy goal for us I think we would have gone on to win it,” he told Sky Sports.
He added on their chances of qualifying for Europe: “We’ve got a long way to go, I’ve always said we’re playing catch up because of the start that we had to the season so we will see were it goes.
“At the moment, today’s result didn’t help them, didn’t help us, doesn’t really close the gap on anybody above us and the games are running out.
“So we missed a big opportunity today at 2-0 up, we should have, I felt, been good enough to go on and win the game.”
