Omonia goalkeeper Francis Uzoho put in an incredible performance for Neil Lennon's side, who were on the verge of picking up their first point in Group E courtesy of a draw in Manchester.
Erik ten Hag's side finally made the breakthrough in the final exchanges, though, with the result keeping them second in Group E, three points behind division leaders Real Sociedad.
Man United, who had 34 shots in total, had the first chance in the third minute when Marcus Rashford collected a short corner before firing one towards goal, but Omonia stopper Uzoho was on hand to make a brilliant save.
Rashford was also involved in the home side's next opening, setting up Cristiano Ronaldo in the ninth minute, but the Portugal international's effort was into the side netting.
Rashford then had a goal-bound strike blocked behind for a corner, before Bruno Fernandes's powerful low drive into the Omonia box evaded everyone in green and red.
Man United should have taken the lead in the 22nd minute when Fernandes found Rashford inside the penalty box, but Uzoho was on hand to make a brilliant save.
Fernandes then curled one just wide of the away side's goal before Uzoho kept out Rashford again, with Omonia surviving another dangerous spell to keep their clean sheet intact heading for the half-hour mark.
Ronaldo fired one wide of the Omonia post in the 32nd minute before Casemiro struck the crossbar from distance, with replays showing that Uzoho had got a touch on the ball.
Fred was next to come close, heading an Antony cross over the crossbar, before Omonia had a brilliant chance of their own on the break, with Bruno firing wide of the post when he might have squared it for Andronikos Kakoullis.
Uzoho made a brilliant double save to deny Antony and then Rashford in the early stages of the second half, with the goalkeeper continuing to keep out the hosts, also denying Fred and Ronaldo during a dominant period for Ten Hag's side.
Luke Shaw set up Fernandes with a low cross just past the hour, but the attacker's effort missed the target, before Uzoho again denied Ronaldo 20 minutes from time.
McTominay was next to test Omonia's remarkable goalkeeper shortly after his introduction, with the Scotland international seeing his low effort beaten away.
Man United continued to push for a winner in the latter stages, and there were five additional minutes added, with Omonia dropping deeper and deeper as the seconds ticked on.
Finally, the breakthrough arrived in stoppage time, with McTominay collecting a deflected Sancho cross before smashing the ball low and hard past the Omonia goalkeeper, who had no answer on this occasion.
Man United - back in Premier League action against Newcastle United on Sunday - will welcome Sheriff Tiraspol in their next European game on October 27, while Omonia will host Real Sociedad on the same night.