Karim Ansarifard sent Neil Lennon's side into a shock lead in the 34th minute of the contest, but the Red Devils scored three in the second period through substitutes Marcus Rashford and Anthony Martial to secure all three points.
Omonia had a second of their own late on through Nikolas Panagiotou, but the 20-time English champions managed to secure all three points at the end of an entertaining contest.
The result has moved Man United onto six points in Group E, but they remain second behind Real Sociedad, who continued their 100% record in the section by winning 2-0 at Sheriff Tiraspol.
Jadon Sancho had Man United's first shot of the match in the sixth minute, but the Englishman's effort was always moving wide of the post, with the visitors dominating the ball early on.
Cristiano Ronaldo - back in the side after being unused in the Manchester derby on Sunday - was next to try his luck for the visitors when he found himself inside the penalty box, but the Portuguese's powerful effort was saved by Omonia goalkeeper Fabiano.
Lisandro Martinez then headed over the crossbar in the 15th minute, before Antony passed up a chance to register when he instead looked for Bruno Fernandes, who was not alive to the situation.
Fernandes hit the woodwork with a lofted effort minutes later, before Ronaldo struck a long-range free kick over the crossbar and Antony tested Fabiano from distance, with the scores still level approaching the 30-minute mark despite the constant pressure from the English side.
Omonia incredibly took the lead in the 34th minute of the contest when a fast break ended with Bruno finding Ansarifard, who expertly picked out the back of the net.
Man United had close to 80% of the ball in the first half, in addition to 12 shots, but they trailed at the interval, with Ansarifard's goal proving to be the difference between the two sides.
Luke Shaw and Rashford replaced Tyrell Malacia and Sancho at the break, and Rashford fired one over the Omonia crossbar inside the first minute of the second period.
Rashford did equalise in the 53rd minute, though, when he brilliantly picked out the far corner of the net after a clever run from Ronaldo had taken the defender away.
Christian Eriksen sent one just wide of the post in the spell that followed, before Antony just missed the far corner, with Omonia coming under increasing pressure.
Man United then scored their second in the 63rd minute when Martial, who had been on the field less than two minutes, collected a flick from Rashford before firing one into the bottom corner.
Bruno had a brilliant chance to level for Omonia 20 minutes from time, but the attacker could not lift the ball over the onrushing David de Gea, with the visitors escaping a very dangerous moment unharmed.
Ronaldo had a golden opportunity to score in the 78th minute when Diogo Dalot found his teammate in a central area, but the 37-year-old somehow hit the post from close range.
The Portuguese turned provider for Rashford in the 84th minute, though, when his effort was turned into the back of the net by the 24-year-old from close range.
Omonia incredibly scored a second in the 86th minute through Panagiotou, but it only proved to be a consolation, with Man United, who had chances to score more late on through Rashford and Ronaldo, able to claim all three points in Cyprus.
Ten Hag's team will now be looking to return to winning ways in the Premier League when they head to Goodison Park on Sunday evening to face Everton.