The defeat in the first leg of their last 16, first leg tie had former Manchester United F.C. captain Roy Keane, now a television pundit, highlighting the quality of the Greek side and saying Moyes needed half a dozen top signings to rebuild his team.
"Criticise Manchester United F.C. all you want but Olympiakos were good. They were on the front foot and technically better than United tonight," Keane, who helped United win the European Cup in 1999, said on ITV.
"There is a lack of confidence and there are some players who just don't have the quality. We've been brainwashed that the Premier League is the best in the world - nonsense," he added.
"It's the best brand in the world but they have fallen behind, United more than most. They need six or seven players to rebuild the club."
Moyes was stunned by the worst performance of a poor season in a competition which remains their only hope of a trophy.
Manchester United F.C. need to turn the tie around at Old Trafford on March 19 because they have only the slimmest chance of returning to the Champions League next season via their Premier league position.
They are 11 points behind fourth-placed arch-rivals Liverpool who occupy the final Champions league qualifying place with 11 matches remaining this season.
"That's the worst we've played in Europe, we didn't deserve to get anything. I'm surprised, I didn't see that level of performance coming," said Moyes days after an encouraging 2-0 win at Crystal Palace in the Premier League.
"I take responsibility, it's my team and we have to play better. There is a second game and we will do all we can to turn it around," added Moyes, who has struggled with the team he inherited from the great Alex Ferguson nine months ago.
"There is undoubtedly talent at Manchester United but we didn't show it tonight," he said. "They have a very good home record so it wasn't unexpected."
Olympiakos are runaway leaders in the Greek league having won all 13 of their home games this season, but United beat them in their four previous meetings when the Old Trafford club were a force to be reckoned with in Europe.
Manchester United F.C. have not reached the Champions League quarter-finals for the last two seasons, having won their third title in 2008 and also reached the 2009 and 2011 finals under Ferguson.
They were undone on Tuesday by goals from two Latin American players, Argentine midfielder Alejandro Dominguez and Costa Rica striker Joel Campbell, neither of whom has been able to hold down a place in bigger teams.
Dominguez was back home two seasons ago on loan from Valencia helping River Plate recover from relegation but was discarded once they regained their top flight status and arrived in Greece via Madrid minnows Rayo Vallecano.
The 21-year-old Campbell, on loan from Arsenal, will relish the prospect of knocking out United as he prepares to face England in the group stage at the World Cup in Brazil in June.