The Toffees dominated the game for the whole of the first half, taking the lead after 10 minutes through Steven Naismith. However, after failing to kill the game off, Adam Le Fondre equalised with a header before securing the win from the penalty spot.
The visitors were also denied two spot-kicks of their own, Martin Atkinson waving appeals firstly when Kaspars Gorkss brought Naismith down inside the area, and then when a Nikica Jelavic shot struck Sean Morrison's arm.
"I think I'm as [angry] as I've ever been after that," he told reporters. "I'm not angry at the referee, he should have given a penalty kick in the first half when it hits [Morrison's] hand. But I'm more angry at my team as we should have scored.
"It's down to us to do it. We didn't punish them and we always left the door slightly ajar for them to come in, and they did do.
"I'm not going to talk too much because I'm not a great lover of the officials. I think certainly one of them is a penalty. I've got no problem with the Seamus Coleman [foul], it's a definite penalty for Reading, so he got that right."
The Toffees have been praised for the quality of their football so far this season, which has seen them challenging for the Champions League places. Moyes insisted they played well in the first 45 minutes at the Madejski, but admitted they need to start putting games to bed.
"The way they played in the first half was outstanding but that's not enough to win the games," he continued. "We're not scoring the goals and when we're getting the opportunities to put the game to bed, we're not taking them.
"In the opening ten minutes of the second half we encouraged pressure and gave the ball away a couple of times in our own half quite simply. We gave them encouragement."
|TOFFEES TO TARTAN|
|50/1||David Moyes is 50/1 with PaddyPower to be the next Scotland manager|
The Merseyside outfit's captain Phil Neville was missing from the match-day squad, and the Scot revealed that his skipper will be out for another six to eight weeks after undergoing knee surgery.
He added: "[Neville] had an operation yesterday on his knee. It'll probably keep him out for 6-8 weeks. His leadership was missed."
The visitors were helped by the Royals' poor display in the first half, a comedy of errors gifting Moyes' men their goal and the hosts creating virtually nothing themselves.
Manager Brian McDermott was delighted that his side recorded their first win of the campaign, while he admitted that the three points, and moving out of the bottom three, could kick-start a turning point in their season.
"Obviously it's a relief," the Royals' boss said. "I'm just so pleased for the players, It's been coming for quite some time. But the only thing that's relevant is the amount of points you've got on the board.
"I said to the players 'forget the result let's just play the game' because everyone has been saying 'you haven't won' and that can play on people's minds. I think [it can be a psychological turning point]."