It was an unhappy return to Goodison Park for the Scot but he was in a positive frame of mind post-game, insisting his side played well in the first half
The Toffees triumphed at Goodison Park to keep up their chase for fourth place, consequently confirming the Red Devils cannot qualify for next year's Champions League for the first time since 1995.
Goals from Leighton Baines and Kevin Mirallas did the damage for Roberto Martinez's side and left Moyes to admit he needs to make changes if he is to guide United back into Europe's premier club competition.
Speaking to Sky Sports he said: "It does [hurt] because it's part of this club. We'll do everything we can to get ourselves in it [Champions League] as quickly as we can. We're going to try to get a team that can win better than this team can."
Phil Jones was at fault for the opening goal, handballing a Romelu Lukaku shot in the box and Moyes was critical of the manner in which his side conceded both goals.
"We gave away two terrible goals," Moyes added. "But we passed brilliantly, well in control of the game but couldn't make enough chances.
"We had great control but were done by two stupid decisions. The first to dive in front of the ball [by Jones] and it was two poor goals to give away.
"We had a lot the ball, passing, and controlled the majority of the game. But we didn't have a cutting edge and got done on the counter attack. We gave away two dreadful goals.
"I've no arguments with the penalty. But it was terrible mistake, because David De Gea would've just picked the ball up."
Despite enjoying the majority of possession, United struggled to create clear chances, with Wayne Rooney particularly disappointing on his return to his former club.
Moyes explained: "We couldn't find the telling pass, or clever play in the final third that makes the difference. I thought we did well, were the better team at 2-0 down in the first half.
"We'll do everything we can to win the remaining four games. We are under way with what we are doing to improve next season and try to give ourselves a better chance to compete at the top end of the table."