The Toffees led with six minutes to play at Carrow Road but conceded twice in the final few minutes to suffer a surprise 2-1 defeat.
Moyes believes that his side controlled the match but paid the price for not adding to Leon Osman's early strike.
He told BBC Sport: "We should have won the game. We were 1-0 up and in control and we let them get back into it and then they scored after the three minutes of injury time [that was scheduled to be played].
"But we didn't do enough to get that second goal."