A chaotic match saw the home side take a 34th-minute lead through Karim Ansarifard, before Rashford and Anthony Martial came off the bench to score three times between them in the second half.
Rashford's second of the match in the 84th minute sent Man United 3-1 ahead, but Omonia struck their second less than a minute later through Nikolas Panagiotou to make it a nervy finish for the Red Devils.
Erik ten Hag's side managed to claim all three points, and the victory has left them second in Group E on six points, three points behind leaders Real Sociedad and three ahead of third-placed Sheriff Tiraspol.
Rashford has conceded that Man United's performance was far from impressive but insisted that securing the victory was always the number one priority.
"Job done, but we would have liked to have played better. We didn't defend badly, but we made mistakes. The first 35 minutes until they scored we were playing well, we just could have got in behind them more," the Englishman told BT Sport after the match.
"Then when he made the subs he [Erik ten Hag] wanted us to be more dynamic, take chances going forward and score goals."
Rashford also said that he prefers to play off the left side of the field but is also happy operating as a central striker, where he has often played in the Premier League this season.
"Left side is probably more natural for me, but I like playing both [left wing and centre forward] and helping the team. If we have got injuries you never know what might happen, where you might be needed to play," he added.
The 24-year-old also paid tribute to teammate Martial, who made it three goals in his last two substitute appearances with a goal in Cyprus.
Rashford's flick set up the Frenchman to make it 2-1 in the 63rd minute, with the striker now appearing to have overcome the injury problems which have restricted his minutes this season.
"He (Martial) is a massive player for us, has been for a long time. When he is happy and at it he is an unbelievable player. We've both been injured, him longer than me, so it is good for him to get minutes and make an impact," Rashford said when questioned on his teammate.
Ten Hag's side will now switch their attention to Sunday's Premier League game against Everton at Goodison Park, with both Rashford and Martial pushing to be involved in the starting XI.