The Red Devils couldn't make their early dominance count as they spurned some good chances before being punished on 34 minutes.
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Erik ten Hag's men were done on the counter-attack as Karim Ansarifard rounded off the move by firing the ball into the roof of the net after Tyrell Malacia was dispossessed.
Just seconds earlier Man United had a free-kick deep in their opponents' half and the goal sparked memories of a goal they conceded in a Champions League defeat at Istanbul Basaksehir in the 2020/21 season.
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Reacting to Omonia's opener, Dean Ashton said on commentary for talkSPORT 2: "What a mistake. How does this even happen when Malacia is the only United player with three Nicosia players around him?
"It's a brilliant finish by the way. He lifts it above David de Gea… incredible.
"It reminds me of a goal they conceded against Istanbul a few years ago when there was no one there to defend."
Ashton later added: "I'm stunned. They were so confident. The goal is shocking where they've allowed a counter attack from your own corner.
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"Then the reaction, they just haven't composed themselves. Omonia Nicosia have grown into it and actually been the better side since the goal.
"If the Manchester derby was embarrassing, this could be more so."
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However, Man United got level in the 53rd minute as Marcus Rashford expertly curled the ball past Fabiano Freitas.
Substitute Anthony Martial turned the match around with a strike from the edge of the box on 62 minutes before Rashford got his second late on 84 minutes from close range.
More slack defending from Man United allowed Nikolas Panayiotou to pull one back for the hosts.
Ashton said: "It's an absolute mess defensively from Manchester United."
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But Man United held on to claim their second win of their Europa League campaign this season. But they were far from convincing…