Alejandro Dominguez opened the scoring with a clever first-half flick and Arsenal loanee Joel Campbell added a sublime second to leave United facing a huge task to overturn the deficit in the return match at Old Trafford next month.
Defeat represented another body blow in an inconsistent season for David Moyes's men.
The Champions League is United's last chance of a trophy and they looked disjointed and sluggish 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 United in five attempts.
- Sports & Recreation
- Joel Campbell
- David Moyes