The Theatre of Dreams turned into the scene of Moyes' worst nightmare on Sunday when United lost 3-0 to Liverpool and questions arose about the manager's future.
But the United boss got everything spot on against the Greeks on Wednesday night.
The odds were against the United boss
The last time the Red Devils came back from 2-0 down in the first leg was 30 years ago when Bryan Robson inspired the team to a rousing 3-0 win over a Barcelona side that included Diego Maradona
Tonight it was the turn of another United icon - Ryan Giggs - to boss the game.
Moyes' decision to deploy the 40-year-old in the middle of the park proved to be a masterstroke as he played a key role in the build-up to the first two goals and excelled throughout.
A battered and bloodied Antonio Valencia, who was given a black eye early in the first half, also put on a display that United's Captain Marvel would have been proud of.
Moyes, who received support from the Old Trafford crowd throughout, may have endured a terrible domestic season, but his European dream is still alive thanks to this impressive comeback.
The only thing that will worry Moyes is that his hat-trick hero was carried off on a stretcher late on with what looked like a knee injury.
United, knowing only a big win would send them through, went at Olympiacos from the off but they were frustrated by the tenacious visitors, who did everything they could to break play up.
Valencia jumped for the ball with Joel Campbell and ended up with a black eye that left him barely able to see
Moyes looked concerned, but the winger continued after treatment.
Kostas Manolas and Alejandro Dominguez saw yellow for petty fouls while Michael Carrick also entered the book for a foul on Hernan Perez.
Perez picked himself up and went down the other end, gaining a yard inside the United box but he fired over
The midfielder fell to the turf and placed his head in his hands
He knew he had missed a gilt-edged chance.
Giggs forgot he was 40 for a moment and his attempt at skinning the Olympiacos right-back ended in embarrassing failure.
But the United veteran showed why he was recalled to the team moments later when he lofted a delicious ball over the top towards Van Persie
Jose Holebas barged over the Dutchman and the referee had no option but to give a penalty.
Van Persie slotted the ball home with 25 minutes on the clock - exactly the same time that Robson had put United ahead against Barcelona in 1984
Could that be an omen, United fans wondered.
Valencia was putting in a battling display in the mould of Robson
The Ecuadorian belted a low shot at goal which almost caught Roberto off guard.
The Olympiacos goalkeeper then saved superbly from Evra as the pressure mounted.
Then, out of the blue, United were under severe pressure
De Gea saved superbly from Perez and he was back on his feet less than a second later to deny David Fuster.
The double-save lifted the crowd and moments later the decibel level inside the stadium rocketed as Van Persie bagged his second of the night.
Giggs, once again, played a key role
The Welshman picked out Rooney with a pin-point pass and the England frontman squared to Van Persie, who tucked away the simplest of side-footed finishes.
Moyes clenched his fists in a rousing celebration.
The Scot knew the job was not complete yet though, and joined his players in jogging away down the tunnel at the break.
Five minutes after the restart United claimed the third goal they needed
Danny Welbeck went down under the faintest of challenges from Manolas but that did not matter to Van Persie.
The striker clipped the ball over the wall and a bamboozled Roberto watched it drift past him
Raucous celebrations ensued, but they soon gave way to a nervous silence as Olympiacos suddenly sprung into life.
Ivan Marcano's deflected free-kick flew a foot wide and Perez missed another good chance, ploughing his low drive straight at De Gea.
The Spaniard saved well again from Dominguez
An injury to Van Persie, who was the victim of a tackle by Giannis Maniatis, meant there were five minutes of added time, but United rode out the storm to make it through to the last eight.
Source : PA
Source: PA