Alejandro Dominguez threatened early on for the hosts after winning the ball in the midfield and driving into the box, before seeing his goalbound effort blocked by Nemanja Vidic.
The Greek side continued to create more of the chances, and Hernan Perez fired narrowly wide of the post with a strike from long range.
In the 38th minute of the match Olympiacos took the lead, as Giannis Maniatis fired in a low strike that was flicked on by Dominguez and diverted into the net beyond the reach of David de Gea.
United almost responded immediately, as Kostas Manolas headed just over his own crossbar from Wayne Rooney’s free kick.
After the break the home side continued to dominate and soon scored their second goal of the evening, when Joel Campbell cut inside from the right and skipped past Michael Carrick, before bending a strike into the bottom left corner.
Olympiacos went in search of a third goal and Maniatis drilled a shot off target from the edge of the box, before Michael Olaitan’s well-struck volley headed just wide of the post.
Robin van Persie should have pulled a goal back for the visitors late on when he was picked out inside the box by Chris Smalling, but after taking a touch and finding space to shoot, he fired over the bar.
It proved to be United’s last chance of the evening, as the home side held on to take a two-goal lead into the second leg at Old Trafford.
Earlier on , Borussia Dortmund easily disposed of Zenit St. Petersburg, 4-2, at the Petrovsky Stadium, getting goals from Marco Reus and Henrikh Mkhitaryan in the first five minutes and later two from Pole Robert Lewandowski.