Campbell has not kicked a ball for the Gunners since signing in 2011, but he showed superb quality to score the second goal in a relatively straightforward 2-0 victory for Olympiacos.
Alejandro Dominguez's clever backheel put the Greeks ahead in the first half and from then on in Moyes' team rarely looked like being able to get back into the first leg of this last-16 tie.
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.
Rooney worked tirelessly, as he always does, and Vidic put in a solid performance at the back.
But United's midfield, as it has done for much of this season, looked awfully pedestrian for the majority of the match, which was played out amid a booming atmosphere inside the Karaiskakis Stadium.
Olympiacos, who are bidding to make the last eight for only the second time, made a mockery of the away side's billing as favourites.
And without the cup-tied Juan Mata and Adnan Januzaj, who was overlooked, United looked toothless in attack.
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 March 19 to make it through to the quarter-finals after this latest sub-par performance.