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.
Olympiacos fans in all four corners of the stadium made their message clear to United from the warm-up: they were not welcome.
The jeering started when the players came out for their pre-match drills and it did not stop for the rest of the night.
Despite the hostile noise being belted out from the stands, United's early showing did not indicate they would buckle under pressure as Olympiacos coach Michel suggested they would on Monday
That was all to change.
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, Olympiacos 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 Olympiacos taking the lead.
What followed was a tepid 20-minute period of play
United always took the safe option on the attack
Olympiacos pushed forward, meaning Rooney spent as much time in his own half as the opposition's.
The only time United entered the home box, Robin Van Persie was out-muscled by Kostas Manolas - a defender Moyes almost signed at Everton.
Tom Cleverley and Giannis 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 David 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 Olympiacos 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 Olympiacos doubled their lead.
The finish - this time struck by Campbell - was another top-class effort.
The Arsenal winger, who has spent time on loan at Real Betis and Lorient, 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 20 yards.
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.
Olympiacos wanted the win more so they were not going to yield as easily as the away side had done earlier.
Source : PA