Alejandro Dominguez and on-loan Arsenal forward Joel Campbell struck either side of half-time for the hosts, who were superior to the Premier League champions for the vast majority of the contest.
United's defence of their top-flight title already looks set to end in failure but, despite also being out of both domestic cup competitions, their performances in Europe have provided a crumb of comfort during Moyes' trying first season at the helm.
But his team were unable to touch the form that saw them through to the last 16 as unbeaten Group A winners.
"It's the worst we've played in Europe, that's for sure - a really poor performance tonight," United manager Moyes said at his post-match press conference.
"We never really got going from the start and we didn't deserve anything because of the way we played.
"We just never really got to grips with things and to a man you could hardly pick anybody out. We just didn't perform tonight.
"We came into the game on good form, with a good mindset, but it didn't show.
"I'm just surprised. I didn't see that level of performance coming.
"I still didn't think we were necessarily two goals worse off in the game but we didn't offer enough on the night to create a goal."
Moyes conceded that the visiting dressing room at the Stadio Georgios Karaiskakis was a bleak place to be at full-time but he retains hope of a famous European triumph to rival any Old Trafford has seen over the club's illustrious history when the return clash takes place on March 19.
He added: "The players are hurting as well, they know how they performed. But it's a team here and we'll stick together.
"There's a second game to come. Old Trafford has seen some great nights in the past and I'm looking forward to hopefully seeing another one."
