Goals from Alejandro Dominguez and Arsenal loanee Joel Campbell gave Olympiacos a 2-0 lead at the halfway point of their Champions League last-16 tie against United.
Given that Moyes' men are 11 points adrift of the top four, winning the Champions League was seen as their best way back into the competition.
But after such a woeful performance in Athens, the odds are firmly stacked against the reigning Barclays Premier League champions making the quarter-finals.
"That's the worst we have played in Europe," the United manager said.
"It was a really poor performance. We never really got going from the start and we didn't deserve anything because of the way we played.
"We never really got to grips with things and to a man you could hardly pick anybody out. We just didn't perform.
"We came into the game in good form and a good mindset but it just didn't show. I am just surprised. I didn't see that level of performance coming. I just didn't see it."
Olympiacos goalkeeper Roberto did not have a save to make all match as Wayne Rooney and his team-mates failed to live up to their pre-match bravado.
Other than a wayward effort from Robin van Persie, United did not look like scoring at all.
It was United's manager who took the blame for the manner of the defeat, though.
"I take responsibility," Moyes said
"It is my time and I will always front it up. We didn't play well. We have to play better."
Although the raucous home support in the sold-out Karaisakis Stadium helped Olympiacos, the Greek champions also showed far more skill and composure on the ball.
United will have to win by three clear goals in the second leg on March 19 to make it through to the quarter-finals, but Moyes has not given up hope just yet.
"The one good thing is that there is still a second game to come," the Scot said.
"We will do everything we can possibly do to reverse the 2-0 defeat."