They could have snatched a draw during the closing moments when substitute James McFadden saw his chip come back off the crossbar, with Andy Johnson unable to stab home the rebound from inside the six-yard box.
"I think the goalkeeper actually pushed it on to the post," said the Everton boss.
"When you look at it, the ball came off the bar and hits [Ulises] de la Cruz on the arm, so it is a penalty kick.
"Andy changes direction then and hits the post. On another day, it hits the inside of the post and it goes in, but it did not do that today."
Despite the defeat, Everton have nevertheless produced a decent opening to the new campaign.
Moyes said: "We still have players to come back, and if you had given me Tottenham on Tuesday night and Reading on a Saturday, coming away and winning one of them, you would probably have said that was a hard thing to do.
"I just think today there was a chance to do more than we did."
Moyes added: "The workrate probably deserved a draw this afternoon, but I do not think the way we played deserved it.
"But you have got to give Reading credit, because they do what they do very well.
"They kept the pressure on and we lost a schoolboy goal from a throw in."