With United stunningly trailing to a first-half goal from Karim Ansarifard, Rashford came on at the interval and scored one and set up another for Anthony Martial to turn the game on its head within 20 minutes of his arrival.
Rashford added United's third with six minutes remaining before Omonia ensured a nervy finale after Nikolas Panagiotou immediately pulled a goal back.
The result leaves United second in Group E on six points, three behind leaders Real Sociedad and three ahead of Sheriff Tiraspol. Omonia, who will visit Old Trafford in a week's time, remain without a point.
1. Substitutes turn it around for Manchester United after a scare
Omonia Nicosia had only won one of their previous 26 European fixtures beyond the qualifying rounds when they welcomed Manchester United to the GSP Stadium on Thursday night but they still managed to give Erik ten Hag's side a scare. Despite dominating the first half in terms of possession and chances, United went in at half-time 1-0 down thanks to a shambolic Tyrell Malacia mistake contributing to Karim Ansarifard's opener after a clinical counter-attack.
Omonia manager Neil Lennon was part of the Celtic team which shocked United in the Champions League group stages in 2006 but his hopes of masterminding another unlikely victory were short-lived as Marcus Rashford and Anthony Martial came off the bench to put the visitors 2-1 up early in the second half. Rashford scored again to make it 3-1 before Nikolaos Panagiotou made it 3-2 late on to set up a nervous finish.
After the demolition in the derby against Manchester City and an early Europa League defeat to Real Sociedad, it wasn't a game Ten Hag could afford to lose and while the result was the right one, the Dutchman won't be happy with parts of the performance. United should have been out of sight well before Omonia scored their first and it was fortunate that Rashford and Martial were on hand to mount a rescue act in the second half.
2. More frustration for Cristiano Ronaldo
It's ironic that the competition Cristiano Ronaldo didn't want to play in this season now offers him the best chance to get a game. Without a start in the Premier League since August, Ronaldo was named in the team for the third Europa League fixture in a row but he endured a frustrating night in Cyprus.
One effort from a tight angle early on was well saved by Fabiano in the Omonia goal before he was penalised for a high boot when he looked sure to score from close range after Bruno Fernandes' lob had bounced off the top of the crossbar. He fired one speculative free-kick high into the stands and then fluffed his best chance of the night when Diogo Dalot squared his pass across the box but Ronaldo saw his effort come back off the post.
He played his part in United's win with a cross to the back post which was turned in by Rashford to make it 3-1 but it's now one goal in nine appearances for the 37-year-old this season and it's unlikely he's done enough to force his way into the team to face Everton
3. Jadon Sancho is struggling for consistency
This was meant to be the season -- his second following his move from Borussia Dortmund in 2021 -- that Jadon Sancho was supposed to show why United were so keen to sign him. But while there have been flashes of his ability with goals against Liverpool, Leicester and Sheriff, he's struggling to turn it on week after week. He was barely in the game at Man City on Sunday and only lasted 45 minutes here.
It was his poor pass which played Malacia into trouble for Omonia's opener and was quickly replaced at half-time with Rashford, who needed only eight minutes to score the equaliser. Ten Hag is a big fan of Sancho, who has started all but one of United's games so far this season but there is an argument that he should sit out the trip to Goodison Park on Sunday.
He's not doing enough to affect games regularly and has a habit of disappearing or taking the easy decision when he gets the ball. Ten Hag has other options in the forward positions and Martial and Rashford, in particular, will feel they have all got a chance to start against Everton.
Omonia Nicosia: Fabiano 7, Miletic 5, Yuste 6, Lang 6, Matthews 6, Lecjaks 6, Diskerud 5, Charalampous 7, Barker 6, Bruno 7, Ansarifard 7.
Subs: Papoulis 6, Kakoulli 6, Loizou 6, Panagiotou 7, Bezus 6.
Manchester United: De Gea 6, Dalot 6, Malacia 5, Lindelof 6, Martinez 7, Casemiro 6, Eriksen 6, Fernandes 6, Antony 6, Sancho 5, Ronaldo 6.
Subs: Rashford 8, Shaw 6, Martial 7, Fred 6, McTominay 6.
Best and worst performers
BEST: Marcus Rashford, Manchester United.
He came off the bench at half-time and changed the game with two goals in the second half.
WORST: Tyrell Malacia, Manchester United.
It hasn't been a good couple of days for the Dutchman, who was substituted at half-time in the Manchester derby and then made the error which led to Omonia taking the lead.
Highlights and notable moments
OMONIA STUN MANCHESTER UNITED. 🤯 pic.twitter.com/wIAWuLWvo3— CBS Sports Golazo ⚽️ (@CBSSportsGolazo) October 6, 2022
Certainly was unexpected.
Marcus Rashford has a brace. 🔥 pic.twitter.com/b6z3Yykl4G— CBS Sports Golazo ⚽️ (@CBSSportsGolazo) October 6, 2022
Marcus Rashford's second score ensured the win for United on a nervy night in Cyprus.
After the match: What the managers, players said
Omonia Nicosia manager Neil Lennon to BT Sport: "I didn't expect us to score two goals! 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."
Manchester United manager Erik ten Hag to BT Sport: "We started well, then an unnecessary mistake and a really bad 10 minutes. We have to learn from it."
Key Stats (provided by ESPN Stats & Information)
- Despite numerous chances on the night to make it 700, Cristiano Ronaldo remains on 699 career club goals in all competitions (450 with Real Madrid, 143 with Man Utd, 101 with Juventus, five with Sporting CP).
Manchester United: Will travel to Everton in Premier League action on Sunday and then another Europa League group stage tilt with Omonia Nicosia on Thursday, this time at Old Trafford.
Omonia Nicosia: A match with AEL Limassol in the Cypriot First Division on Monday before Thursday's trip to Manchester.