Alejandro Dominguez opened the scoring with a clever first-half flick and Arsenal loanee Joel Campbell added a sublime second goal to leave United facing a huge task to overturn the deficit in the return match at Old Trafford on March 19.
"That's the worst we've played in Europe. We didn't deserve to get anything," Manchester United F.C. manager David Moyes told reporters. "I'm surprised, I didn't see that level of performance coming.
"I take responsibility, we have to play better. The players are hurting."
Defeat at the Georgios Karaiskakis Stadium represented yet another body blow in an inauspicious campaign for last season's English Premier League title winners.
Manchester United F.C. are out of the FA Cup and the League Cup and trail Premier League leaders Chelsea by 15 points with 11 games left to play.
Moyes's men looked sluggish and disjointed against an inexperienced Olympiakos side who are bidding to reach the quarter-finals for only the second time in their history.
It was the first time the Greeks had beaten Manchester United F.C. in five attempts.
"To beat a great side like United, the way we did, is something very special and I can't hide the joy and pride I feel for my players," said Olympiakos coach Michel.
"We picked the best moment to show Greek football is alive and kicking," added the former Spain and Real Madrid striker.
Moyes opted to start with two wingers, Antonio Valencia and Ashley Young, but neither made an impact in a first half in which runaway the Greek league leaders displayed real energy and hunger.
After a cautious opening from both sides amid ear-splitting noise from an enthusiastic home crowd, the first chance fell to Olympiakos as Dominguez burst through the middle on a solo run only to be denied by a last-ditch tackle from Nemanja Vidic.
Midfielder Tom Cleverley registered United's only effort of note in the first half with a 17th-minute half-volley that sailed high over the bar.
Strike pair Wayne Rooney and Robin van Persie were peripheral figures for the visitors as pot shots on goal from Giannis Maniatis and Hernan Perez lifted the home supporters.
Olympiakos, wearing bright red and white jerseys, red shorts and socks, broke the deadlock seven minutes before halftime with a smart piece of improvisation from Argentine forward Dominguez.
Patrice Evra's half-hearted clearance from a corner landed at the feet of Maniatis 25 metres out and his tame, low shot was expertly diverted past keeper David De Gea by Dominguez's flick.
United, the European champions in 1968, 1999 and 2008, were punished again for poor defending 10 minutes into the second half.
Campbell, who is likely to feature for Costa Rica against Group D rivals England at the World Cup in Brazil in June, did the damage.
He picked up possession 25 metres out, jinked past midfielder Michael Carrick and sent a curling left-foot shot round defender Rio Ferdinand and into the bottom corner of the net.
Moyes sent on Danny Welbeck and Shinji Kagawa for Valencia and Cleverley but it was more a change in attitude than personnel that was required from United.
Dutch striker Van Persie summed up his team's night with five minutes to go. He received the ball 12 metres out and turned inside a defender before blazing a right-foot shot way over the bar.
"It is a great moment for us and we will enjoy it but we must look ahead to the second leg. Whoever thinks 2-0 is enough to qualify is mistaken," said Michel.
"We expect a different United at Old Trafford. We have huge respect for them so we have to keep working if we want to write another piece of history."