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.
Rodriguez opened the scoring in the 38th minute when his clever flick just inside the box deflected a long-range shot past keeper David De Gea.
Campbell put the game beyond United 10 minutes into the second half with a brilliant curling left-foot effort from 20 metres.
Robin van Persie had United's best chance in the closing stages but he ballooned the ball way over the bar.
