Anthony Martial also scored while Karim Ansarifard and Nikolas Panagiotou netted for the home side as United claimed a narrow win on their first ever trip to play a Cypriot team.
On a balmy night at Nicosia's 23,000 capacity GSP Stadium, United went a goal down in the first half but fought back for a morale-boosting victory after being demolished 6-3 by cross-town rivals Manchester City at the weekend.
Erik ten Hag named a strong line-up with Cristiano Ronaldo, starting just his fourth match of the season, leading the attack and Brazilian Casemiro anchoring the midfield, but it was the substitutes who saved the day.
"All the subs had a really good impact", with (Luke) Shaw and Rashford strengthening the left side," Dutch coach ten Hag told reporters afterwards.
"We played well, the only thing is we didn't score a goal. After half-time I think we dealt with the setback.
"It was a bad 10 minutes after we conceded the goal (but) if you then are able to come back... that's positive."
United got off to a dominant start, largely controlling the opening period against a side managed by former Celtic boss Neil Lennon.
Antony nearly got the breakthrough in the 28th minute, cutting inside and unleashing a long-range shot that was parried away by Omonia goalkeeper Fabiano.
It looked like United were beginning to turn the screw when they were awarded a free-kick outside Omonia's area in the 34th minute.
Christian Eriksen fired it in but Omonia managed to scramble it away. The ball fell to Tyrell Malacia who was then robbed by Omonia's Brazilian forward Bruno.
Three Omonia players broke clear before Bruno tapped it across the face of goal for Iranian striker Ansarifard to tuck it away.
United came out stronger in the second half with Rashford coming on for the anonymous Jadon Sancho and Shaw replacing Malacia.
And it paid immediate dividends, with Bruno Fernandes getting a shot on goal before setting up Rashford to level the scores in the 53rd minute.
Ronaldo, who was left on the bench during the weekend rout by City, cut a frustrated figure up front, but United continued to drive forward, creating more chances as the second half progressed.
Ten minutes after levelling, United went ahead when the newly-introduced Martial tapped in with his second touch of the game, seizing on an assist from Rashford.
The home supporters jeered loudly when referee Joao Pinheiro of Portugal waved away an Omonia claim for handball, a decision confirmed by VAR.
Ronaldo's woes continued, however, with the Portuguese star hitting the post from within the six-yard box with an open goal, one of a hatful of missed chances by the 37-year-old.
Minutes later Rashford latched onto a pass from Ronaldo to double his tally.
Omonia were not finished, though, Panagiotou getting a goal back in the 87th minute.
United created more chances, with Brazilian midfielder Fred forcing a save from Fabiano in the dying minutes.
Lennon praised his players for putting up a fight but said he saw signs that United were on the up.
"This is a team definitely building... I think they're going in the right direction," he said.
"Thankfully Cristiano didn't have his shooting boots on tonight."