Out of both domestic cup competitions and comfortably adrift in the Premier League title race, United manager David Moyes spoke boldly this week of claiming Europe's top prize.
The message did not appear to get through to his players on a night of toil in Piraeus.
An instinctive 38th-minute flick from the excellent Dominguez gave Olympiacos a deserved lead before on-loan Arsenal forward Campbell ensured they will protect an two-goal cushion at Old Trafford on March 19.
The visitors badly lacked a creative spark, with Moyes' decision to employ Antonio Valencia and Ashley Young in the wide midfield positions, while leaving Adnan Januzaj and Shinji Kagawa on the bench sure to come under the spotlight.
Nevertheless, the embattled former Everton boss should have had his blushes partially spared when Robin van Persie passed up a glorious late chance.
Top scorer Kostas Mitroglou's January departure to Fulham and a thigh injury suffered by seasoned forward Javier Saviola left 21-year-old Nigerian Michael Olaitan to lead the Olympiacos attack.
Moyes' men made a nervy start amid a fervent atmosphere in Stadio Georgios Karaiskakis - Rio Ferdinand twice losing possession in a matter of seconds before Dominguez's resulting shot was blocked.
Dominguez was Olympiacos' leading light in the opening exchanges and Nemanja Vidic was forced to make a last-ditch tackle to thwart the playmaker's storming eighth-minute solo run.
United belatedly settle and Tom Cleverley might have taken a 17th-minute chance, slashing over when the host failed to clear a corner.
Hernan Perez whipped a 26th-minute shot against the side netting and Olympiacos continued to carry the greater threat before going ahead in fortunate circumstance.
Campbell's cross was cleared as far as Giannis Maniatis and, although the midfielder's speculative effort appeared harmless, Dominguez was alert enough inside a crowded penalty area to divert the ball past David de Gea.
United almost found an instant response, Kostas Manolas heading Wayne Rooney's excellent free-kick from the left narrowly over his own bar, but an aimless pass into touch from Cleverley 30 seconds into the second half suggested their struggles were set to continue.
And so it proved in the 54th minute when Campbell greeted Michael Carrick's half-hearted attempt at a challenge with a nutmeg before curling a measured shot beyond De Gea from 25 yards.
With the tie slipping away, Moyes introduced Kagawa but Giannis Maniatis went close to adding an improbable third, chesting down and firing over after Carrick failed to clear.
Olympiacos began to drop dangerously deep in the closing stages and United should have punished them with a vital away goal in the 81st minute, but Van Persie blasted off target having collected Chris Smalling's cross.
