Firmly against the run of play, Karim Ansarifard sent the Cypriot outfit into the lead against the 20-time English champions in the 34th minute of the contest.
Marcus Rashford scored twice and also set up Anthony Martial in the second half, as Man United turned the match around, but Omonia found the back of the net on a second occasion in the 85th minute through Nikolas Panagiotou.
Lennon has said that the home team had to 'ride their luck', with Man United having 27 attempts on goal and missing a number of opportunities throughout the match.
However, the Scot is confident that his side can cause the Red Devils problems in the reverse match at Old Trafford next week after identifying some "weaknesses".
"We rode our luck a lot of times, we kept ourselves in the game with good goalkeeping and a lot of Man Utd missed chances. I didn't expect us to score two goals!" Lennon told BT Sport.
"It's going to be a different game at Old Trafford, but we have seen them up close and know what their weaknesses are. We will go there and give it everything we've got."
Lennon also said "some of the players" would have regrets looking back at the match, with Bruno missing an excellent chance in the second half of the contest.
"Some of the players will, yeah (will have regrets). The first two goals we were too open at centre-half then Bruno misses the chance with the lob. There were things we could have done better with ball retention. The gulf between the teams was there, but I'm delighted with two goals," he added.
Omonia are at the bottom of Group E, having lost all three of their matches in the section to Sheriff Tiraspol, Real Sociedad and now Man United.
The Cypriot side will travel to Old Trafford, host Real Sociedad and visit Sheriff in their final three matches in the group, and they can still mathematically qualify for the next round.
Omonia have won their last two league matches against Enosis and Doxa but are currently down in fourth position in their domestic table, collecting nine points from five matches.
Lennon has been in charge of the club since March 2022 and led them to the Cypriot Cup last season.
The 51-year-old has had two spells in charge of Scottish giants Celtic, while he has also managed Bolton Wanderers and Hibernian since retiring as a player in 2008.