United showed more urgency as the game wore on, but lacked potency in attack and David Moyes will want a big improvement in the second leg at Old Trafford in three weeks.
For most of the first half, both sides struggled to fashion any goalscoring chances of note. Nemanja Vidic made a superb early block to deny Dominguez as he was about to pull the trigger and it took 17 minutes for United to fashion a first half-chance, but Tom Cleverley could only skew a volley over the bar following a corner.
The game was largely being played in the middle third but, as seems to have happened often this season, the Reds fell behind in unfortunate circumstances in the 38th minute as Dominguez diverted Giannis Maniatis' shot past a stranded David De Gea.
United came out after the break with more purpose but were undone again when Campbell bent an excellent shot past an unsighted De Gea. Moyes brought on Danny Welbeck and Shinji Kagawa in a bid to inject some life into his team but home keeper Roberto remained untroubled and Michael Olaitan came within inches of scoring Olympiacos’ third.
United suddenly sprung into life with 10 minutes left and Robin van Persie missed a golden opportunity to net a vital away goal, blasting over from 12 yards after controlling Chris Smalling’s cross. It summed up a miserable evening but the Reds will have another bite at the cherry on 19 March.