The Toffees fell behind at Old Trafford after 13 minutes when Ryan Giggs finished neatly from close range.
Moyes believes that Everton responded well after going behind, but admits that his side's defending was not up to the standard required to trouble United.
He told BBC Sport: "We gave Manchester United an early start but I thought they were great. We didn't come out of the blocks as fast as we would have liked to have done.
"The second goal was poor, I don't know why we were so high and square at the back. But the timing of it was the worst thing.
"I thought we played well after the opening goal and had a grasp of the game. Manchester United are always a threat when you don't make enough of your opportunities. We didn't have that cutting edge today."
Defeat at Old Trafford means that Everton stay sixth in the Premier League, six points off a Champions League place.