Moyes bemoans 'terrible' goals
The Scot suffered a miserable first return to Goodison Park, as a Leighton Baines penalty and Kevin Mirallas' narrow strike handed Everton a comfortable victory.
Baines tucked away the 28th-minute spot-kick after Phil Jones had inexplicably handled Romelu Lukaku's shot, and Moyes was left distinctly unimpressed by his side's defensive showing.
"We gave away two terrible goals," he told Sky Sports.
"We get done by two stupid decisions - well, a stupid decision to dive in front of the ball in one of them.
"Prior to that we had passed the ball well, kept it well, and were in control of the game.
"We played well in the first half, but didn't have a final through-pass or a final finish, that was the disappointing part."
The defeat means it is now mathematically impossible for United to reach the top four and a spot in next season's UEFA Champions League.
Moyes confirmed the blow would be painful to take, but insisted work is going on "behind the scenes" to improve his squad ahead of next season.
"It does (hurt), because it's part of this club and we want to be in it," he said.
"We want to get back in it as quickly as we can.
"We're going to try to get a team that can win better than this team can."