Having entered the game 2-0 down on aggregate, United needed to win by three clear goals to book their place in the last eight and avoid a surprise exit at the hands of the newly-crowned Greek champions.
Robin van Persie got them on their way with a first-half penalty before adding a second right on the stroke of half time and completing his hat-trick with a free kick after the break.
Here, Sports Mole takes an in-depth look at a memorable European night at Old Trafford.
On target: 5
On target: 3
Was the result fair?
In truth, a 3-0 win is harsh on Olympiacos. The stats and figures suggest that it was an even game and, for large periods, it was exactly that. The visitors played their part and certainly had enough chances to get what would have been a winning away goal.
United certainly did not dominate proceedings, but they had an urgency about them that always made them look most likely to score. The fact that they found the back of the net three times is no surprise, but the fact that they kept a clean sheet is.
Over the whole tie, Olympiacos will feel aggrieved to have gone out as they were a match for United in both games. However, it was their failure to convert their chances that cost them tonight, and United took full advantage.
Man Utd's performance
Much, much better. United fans were expecting - demanding, even - a reaction to their humiliation at the hands of Liverpool on Sunday, and they certainly got it. There was more urgency, pace and passion about United's play than there has been for a long time.
The likes of Wayne Rooney and Robin van Persie have been criticised for their form recently, but they turned up tonight and were supported by an able cast including a fine performance from the 40-year-old Ryan Giggs, who ran the show from midfield at times.
They did look shaky in defence, but every time that Olympiacos managed to get through, they found David de Gea in superb form. The Spaniard kept the visitors out numerous times, including a world-class double save in the first half.
The match was rightly viewed as a huge one for the club, and the celebrations at the final whistle reflected that, but it is important to remember that they were expected to overcome Olympiacos when the draw was first made, and tonight's result really should be seen as the bare minimum of achievement. They now need to kick on from tonight's game and make sure that it acts as a turning point in their season.
Michel will be scratching his head and wondering quite how his side lost this match 3-0. They were dangerous in attack and were not exactly calamitous in defence - although two mistakes did lead to goals.
The visitors deserve plenty of respect for refusing to sit back on their lead and not parking the bus. They went into the game looking for a crucial away goal when plenty of other sides would have simply looked to stop United playing and crowded out their own box.
They were United's equal for large periods of the match and would have felt relatively in control and confident, even at 2-0. However, a mixture of inspired goalkeeping and poor finishing meant that they couldn't find the all-important goal and suffered a disappointing exit from the competition.
Sports Mole's man of the match
Robin van Persie: A number of United players are worthy of winning this award tonight, be it Rooney for his tenacious and inspirational display, Giggs for his timeless poise and creativity or De Gea for his crucial double save.
However, it goes to the hat-trick hero Robin van Persie, who proved his recent critics wrong with his first ever three-goal haul in the Champions League. He won the penalty before dispatching it for his first, while the second was a smart finish at a crucial time in the match.
He got slightly lucky with his third as it wasn't the best free kick he would have ever hit and the keeper should have done a lot better, but he still put it in the right place and celebrated a memorable and match-winning hat-trick.
There are two contenders for this one. The first goes to Jose Holebas, who conceded the penalty when he bundled into Van Persie inside the box. It was a silly challenge from the Greek full-back as he was never going to get the ball, and it resulted in the goal that sent United on their way nice and early.
The second is for Roberto Jimenez in goal as he remained rooted to the spot for Van Persie's free kick despite the ball before just a couple of yards away from him. He made the slightest move to his left - perhaps put off by Rooney's fake shot - which wrong-footed him for Van Persie's effort, and he could not recover in time.
Bjorn Kuipers had a busy game tonight, but he managed it well. The biggest decision for him to make was the penalty, but he had little choice but to point to the spot. It was a clumsy challenge from Holebas, but Kuipers was also right not to send the full-back off.
He produced a flurry of yellow cards early on which could have set a dangerous precedent, but things soon calmed down and he enjoyed a solid performance all told.
Man Utd: Next up for David Moyes's side is a Premier League clash away to West Ham United on Saturday.
Olympiacos: That is almost the season over for Olympiacos. Their next match sees them take on Ergotelis in the league but, having already been crowned champions and eliminated from all other competitions, it is simply a case of seeing the season out now.