Argentine Alejandro Domiguez gave the host the lead when he redirected Giannis Maniatis's shot past David de Gea on 38 minutes.
Joel Campbell - on loan from Arsenal - doubled the advantage for the Greek champion on 55 minutes with a superb curling left-foot strike from the edge of the box after a nutmeg to get past Michael Carrick.
Wayne Rooney and Nemanja Vidic had talked up United's chances of winning the Champions League in the build-up to the match, but it was easy to see why David Moyes's side was a 25-1 shot to land the trophy going into this match in the port suburb of the Greek capital.
Rooney worked tirelessly, as he always does, and Vidic put in a solid performance at the back.
But United's midfield, as it has done for much of this season, looked awfully pedestrian for the majority of the match, which was played out amid a booming atmosphere inside the Karaiskakis Stadium.
Olympiacos, which is bidding to make the last eight for only the second time, made a mockery of the away side's billing as favourite.
And without the cup-tied Juan Mata and Adnan Januzaj, who was overlooked, United looked toothless in attack.
With United 11 points adrift of the top four in the Premier League, winning the Champions League may be the best way to get back into the competition next year, but it will have to win by three clear goals in the second leg on 19 March to make it through to the quarter-finals after this latest sub-par performance.
