Moyes endured an unhappy first meeting with the club he left in the summer as Bryan Oviedo's 86th-minute winner secured Everton's first Premier League success at the home of the champions in 21 attempts.
And, having conceded that his side could not afford any more slip-ups in the wake of the 2-2 draw at Tottenham on Sunday, Moyes now knows the task of defending their crown has become much more difficult.
With Arsenal now 12 points ahead at the top of the table and with Chelsea, Manchester City, Liverpool and Tottenham all triumphing on Wednesday night the Red Devils currently sit ninth, five adrift of the UEFA Champions League places.
And when it was put to him his side now have significant ground to make up on their rivals, Moyes told Sky Sports: "Yes, big catch-up."
He added: "We are very disappointed. We didn't take a couple of chances that we had - we needed a bit of good fortune, but one or two things hit the post and Tim Howard made one or two saves.
"Everton kept at it and showed how good a team they are. They got the one goal.
"We had opportunities, but Everton did as well, they've been playing well and that's why they're up near the top of the league."
The only goal came in the closing stages as Romelu Lukaku's cross-shot was turned home at the far post by an unmarked Oviedo, while Moyes admitted he felt the build-up to the game should have focused on the players, and less on his meeting with his old club.
He said: "We didn't defend it well enough and we could have cleared it. We gave the ball away a minute before which led to the goal, so it was our own doing.
"It should have been about the players, about the players at Everton and how well they are doing.
"I didn't think it was necessarily about me at all, I was just fortunate to manage Everton and I'm fortunate to be managing Manchester United."