The Red Devils never got going in their last-16 encounter and fell behind seven minutes before the interval when Alejandro Dominguez deflected an ambitious effort from Giannis Maniatis past a wrong-footed David de Gea.
Things went from bad to worse for David Moyes' men early in the second half, with Joel Campbell -on loan at Olympiakos from Arsenal - working the ball onto his left foot 30 yards out and curling brilliantly into the bottom corner.
United did create one late opportunity to salvage something from a forgettable performance, but Robin van Persie lashed over the top with his right foot and his team-mates have it all to do if they are to remain in contention for the one piece of silverware still available to them this season.
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 Moyes' side were 25-1 shots to land the trophy going into this match in the port suburb of the Greek capital.
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 they will have to win by three clear goals in the second leg on 19th March to make it through to the quarter-finals after this latest sub-par performance.
Patrice Evra went into the book in the second minute, when he handled Campbell's cross. But just when United started to settle into a rhythm, Olympiakos opened them up.
Chris Smalling slipped and the ball made its way to Dominguez, who ran through on goal. Only a superb sliding tackle from Vidic stopped the hosts from taking the lead.
What followed was a tepid 20-minute period of play. United always took the safe option on the attack. Olympiakos pushed forward, meaning Rooney spent as much time in his own half as the opposition's.
The only time United entered the home box, Van Persie was out-muscled by Kostas Manolas - a defender Moyes almost signed at Everton.
Tom Cleverley and Maniatis both had weak shots at either end of the pitch, but the overall quality of the game was poor.
The home crowd remained almost deafeningly loud, however, and the decibel level rose again as Hernan Perez cut in from the left flank and fired a powerful drive which a relieved De Gea watched hit the side-netting.
The energetic Dominguez blotted his copy book by play-acting while trying to win a penalty, but he soon redeemed himself in the eyes of the noisy home supporters when he cleverly put Olympiakos ahead.
Maniatis took advantage of Evra's weak clearance, driving at the United defence before unleashing a low shot which Dominguez diverted past De Gea with a clever backheel.
United hung on until half-time, but nine minutes after the break their troubles worsened as Olympiakos doubled their lead.
The finish - this time struck by Campbell - was another top-class effort.
The Arsenal winger just about kept his balance after nutmegging Michael Carrick and he then curled a stunning shot around Rio Ferdinand and the sprawling De Gea from distance.
The man on the loud hailer behind the home goal bellowed and the home fans demanded their team go in for the kill. Michael Olaitan nearly obliged, firing a 20-yard shot just wide of a flat-footed De Gea.
Campbell received a standing ovation when he came off moments later.
Moyes brought on Shinji Kagawa and Danny Welbeck but they could not make the difference.
United barely threatened thereafter. Van Persie slammed a shot into the crowd from inside the box, but that was about it.
Olympiakos wanted the win more so they were not going to yield as easily as the away side had done earlier.