Moyes described the match as "the worst performance in Europe" he'd seen from his players, and took full responsibility for the result.
Argentine striker Alejandro Dominguez broke the deadlock at Karaiskaki Stadium when he got his foot to a shot from Yiannis Maniatis in the 38th minute and turned the ball past goalkeeper David de Gea.
First-half composure from the visitors soon turned to complacency after the break when striker Joel Campbell, on loan from United's Premier League rival Arsenal, caught De Gea off guard with a well-placed shot in the 55th minute.
Wayne Rooney produced no threat up front and strike partner Robin van Persie missed the visitors' only clear chance — a shot over the crossbar in the 82nd minute — as United slipped to its first defeat in the Champions League this season.
"It was a very poor performance, and we didn't deserve to get anything. To a man there was nothing," Moyes said.
"I'm just surprised. I just didn't see that level of performance coming. We didn't offer enough on the night to create any goals."
Olympiakos coach Michel Gonzalez played down the implications of the result, saying that 2-0 wasn't a big enough margin to give an advantage in terms of qualifying.
"I will never say anything bad about Manchester United, because I have such great respect for them. The players know that we still have more to do," Gonzalez said.
"But I can't hide it. I am very happy. It's the players I'm so proud of ... to get this result against a team of this calibre."
Runaway leaders in the Greek league, Olympiakos fielded its untested strikers after the sale last month of Kostas Mitroglou. But the hosts threatened early with Dominguez nearly grabbing a goal in the eighth minute, stopped only by a last-ditch tackle from Nemanja Vidic.
Moyes brought on Danny Wellbeck and Shinji Kagawa in the second half but they made little impact, while Olympiakos nearly made it 3-0 in 67th but Michael Olaitan failed to finish.
Despite the defeat, United midfielder Michael Carrick insisted that the three-time European champions could turn the result around in the second leg on March 19 at Old Trafford.
"We just haven't been getting the right results," Carrick said. "As players we have to stand up ... that's the business we're in."
Premier League champions last season under Alex Ferguson, United is currently sixth in the domestic standings — 15 points adrift of leader Chelsea. It also made an early exit from the FA Cup at the hands of Swansea.