Erik ten Hag's Red Devils made the 2,100-mile trip for Thursday's Group E encounter reeling from Sunday's chastening 6-3 derby defeat at rivals Manchester City.
The United boss thanked Pep Guardiola's side for the "reality check" ahead of the trip to Cyprus, where Omonia threatened an even greater shock after Karim Ansarifard struck a dream opener for the hosts.
But ex-Celtic boss Lennon could not celebrate a famous win as Rashford's brace and a strike from Martial set up a comeback victory that ended 3-2 thanks to Nikolas Panayiotou's late goal.
The victory in Nicosia kept United's knockout ambitions on track and was a result few saw beyond given Cristiano Ronaldo and Casemiro came into a surprisingly strong line-up.
It looked a matter of when rather than if the visitors would score during a one-sided opening period, only for Tyrell Malacia to be dispossessed and Bruno to lead a break that ended with Ansarifard firing home.
Ten Hag made changes in a bid to turn things around at GSP Stadium and they worked out brilliantly.
Half-time introduction Rashford produced an excellent finish to draw level and Martial then scored two minutes after replacing Bruno Fernandes with a fine strike of his own.
Ronaldo hit a post as he looked for his 700th goal but soon his cross-shot was turned in by Rashford, with Panayiotou pulling one back as United held on for victory on a surprisingly challenging night.
United now turn their attention to the league trip to Everton before next Thursday hosting Omonia, who included Wales international Adam Matthews and ex-City prospect Brandon Barker in their starting line-up.
Lennon's hosts were on the backfoot from the outset, with goalkeeper Fabiano denying a Ronaldo snapshot.
Antony should have rounded the Omonia goalkeeper or produced a better pass for Fernandes as United continued in the ascendancy, with the latter seeing an effort loop onto the crossbar soon after.
Fabiano produced an exceptional one-handed save to prevent Antony superbly curling home as Omonia struggled, only to capitalise on lax defending and land a stunning blow in the 34th minute.
With players up to attack a Christian Eriksen free-kick, Jadon Sancho played back to Malacia when the initial cross was cleared and the left-back was all too easily dispossessed by Bruno.
The Brazilian led a rapid Omonia counter and coolly played in Ansarifard to superbly lash past David De Gea.
It was a brilliant break that left United rocking for the remainder of the first half.
Luke Shaw came on for Malacia at the break and Sancho was replaced by Rashford, who scored within eight minutes of his introduction.
Fernandes sent a long ball forward to Rashford, who recovered from a poor first touch to move into space freed up by a smart Ronaldo run and hit a low curling shot from the edge of the box.
Eriksen, Antony and Ronaldo had efforts as United pushed to take the lead, with Martial replacing Fernandes - another alteration that paid immediate rewards.
Two minutes after his 61st-minute introduction, the France international collected a lovely Rashford flick and continued to hit a low strike past Fabiano.
There was a scare when Bruno was put through but the Omonia player just failed to make a clean connection with the ball when trying to lift over De Gea.
Ronaldo wasted a glorious chance to put the game to bed when Diogo Dalot kept his cool under pressure from Fabiano to square for his compatriot, who somehow hit the near post with the goal at his mercy.
The 37-year-old's frustration at not scoring was clear, as was his happiness when Rashford turned home his cross-shot at the back post in the 84th minute.
But Omonia ensured it would be a frantic ending a minute later as Victor Lindelof was caught out and Panayiotou fired home fellow substitute Andronikos Kakoullis' cross.
Fred forced Fabiano into action from distance and Ronaldo struck wide in stoppage time as the clock wound down.