Theo Walcott's deflected strike gave the Gunners the lead after just 50 seconds at Goodison Park, however, Marouane Fellaini's smart finish from the edge of the box drew the Toffees level midway through the first half.
Everton felt they should have had a penalty when Mikel Arteta and Steven Pienaar clashed in the box, and Moyes felt his team did not get the rub of the green when it came to the decisions made by referee Michael Oliver.
"It was a really difficult night for us with the officials," he told Sky Sports. "We got very few decisions tonight.
"If you are going to win games, now and then you need a bit going for you. A lot of results hinge on what referees do and the decisions they make.
"If we were going to win it was going to be without any decisions from the officials going in Everton's favour that's for sure."
Despite his obvious frustration Moyes was quick to praise his team for their efforts, including goalscorer Fellaini, who was back in the side following suspension.
"I thought he [Fellaini] was fabulous tonight," he added. "But our whole team played well, as we had to do against Arsenal. Overall I was really pleased with how the players played.
"Arsenal played well, they had some moments but great credit to our lads after going a goal down after 50 seconds to get right back in it.
"I thought we deserved it when we scored, we were playing well, and it was a really good goal by Marouane."