The Toffees had a Sylvain Distin goal ruled out in a tense clash as the two sides cancelled each other out in a frenetic but goalless clash.
Moyes' side are now five points behind Chelsea and Tottenham, who occupy fourth and fifth positions respectively with two games of the season remaining.
Everton have a home game against West Ham to come next Sunday before finishing their campaign off at Stamford Bridge.
But Moyes was left upset with Distin's effort, which was disallowed by referee Michael Oliver for a foul in the area.
"It was a goal," the Scot told Sky Sports. "Their goalkeeper runs into our player so the goal should stand. I can understand if our player ran into their keeper, you can give a foul for that, but the goalie ran into him, so it was a goal.
"We're still in there (for a European spot). We picked up a point today. We had a really difficult run it, which everybody kept reminding me.
"We've got two games to go and we'll keep pushing to try and make Europe if we can."