It was an epic end-to-end affair that resulted in the Red Devils taking a two-goal lead in the first half, but the Greek outfit continually presented a threat on the counter-attack.
Robin van Persie bagged a hat-trick when it counted the most, and United's performance was—at least in parts—much-improved compared to recent weeks.
Let's take a tactical look at the game and see how it played out.
Formations and XIs
Manchester United lined up in a very fluid 4-4-1-1 shape, with van Persie up front and Wayne Rooney drifting in and around the hole. Ryan Giggs played as a central midfielder alongside Michael Carrick.
Olympiakos mirrored the shape, with Joel Campbell as a lone frontman and Chori Dominguez in behind. Delvin N'Dinga anchored the midfield.
Olympiakos produced a similar brutish, forceful game plan to the one used in the home leg, but they weren't quite so aggressive in their positioning and pressing.
Still, the midfield worked hard to win second balls and spring quick counter-attacks, with pace in the wide areas and up front key in moving the ball quickly.
Either Dominguez, Hernan Perez or David Fuster would carry the ball quickly from deeper areas or feed Campbell early in the forward line.
The Costa Rica star continually dropped into great positions to pick the ball up, turn and run at Manchester United's defence and worry it. In many ways, it received a similar treatment to the one Raheem Sterling gave it at the weekend.
There was space to be exploited either behind Phil Jones or in front of him as the defensive line dropped off, and United's lack of a true anchor in midfield meant the runners were squeezing through unchallenged.
The addition of Giggs to central midfield caused a few gasps pre-match, but his presence in the heart of all things United was vital to their resurgent display.
The only reason the game was end-to-end was because United matched Olympiakos' furious pace, moving the ball quickly from player to player and line to line.
This is what United have been needing to do for months, and Giggs' sharp, accurate passing after taking just a single touch was instrumental.
Like Campbell, Rooney was frequently dropping into holes in and around N'Dinga to receive longer passes from either Rio Ferdinand or Giggs. He picked up an assist and completed six dribbles—a real creative force outside the box.
Strategically, Olympiakos set themselves up well for an aggressive, disruptive game plan.
But their confrontational style took a hit when two of their back four were yellow-carded early on, and United's wingers began finding room to dribble and explore against more hesitant, careful full-backs.
As soon as the Thrylos took their pedal off the gas, United struck the post via a Rooney header. It wasn't long before the Red Devils then took the lead from the spot, with RvP baiting Jose Holebas into a silly challenge.
Rooney then fired a low cross in from the right for his striker to slot home just before half-time, and it was clear Michel's game plan was unravelling before our very eyes.
Olympiakos' Subtle Switch
The Greek champions were utilising a 4-4-1-1-esque shape in the first half, but switched for the second half and employed a more obvious 4-1-4-1. Hardly the most noticeable or drastic change, but the intention was very clear.
Dominguez dropped in alongside Giannis Maniatis in central midfield, and N'Dinga played a very Etienne Capoue-ish role as a sole anchor between the lines.
The key for Michel here was first- and second-ball pressure; a weakness was spotted in Carrick and Giggs' lack of physicality, and if they could be dispossessed early in United's moves, Campbell could be released immediately on rapier-like counters.
The midfield four fought hard for the scraps and provided several chances for their striker to steam forward, but decision-making let them down in the final third.
Campbell fashioned two excellent chances by drifting wide in behind Patrice Evra and hitting the byline to cross, but his midfield colleagues failed to find a way past the superlative David de Gea.
After 60 minutes at 3-0, United had in their possession the bare minimum to qualify for the quarter-finals.
The shocking thing was David Moyes knew it, and the team decided to sit back—either at his command or by reflex—drop the intensity, stop pressing and hold on.
The Old Trafford crowd roared their disapproval, knowing that one goal for the Greek visitors would send the Red Devils packing. Olympiakos began to see more of the ball, fashioned an additional four chances on goal quickly and looked a threat in possession for the first time.
United conceded all the space the away side could ever hope for on a Champions League night, and despite an excellent 60-minute showing, ugly questions reared their ugly heads late on.
Why did they pack up and hold on?
An improved showing from United, and for the first time in months, RvP and Rooney combined to wonderful effect.
The catalyst was Giggs in central midfield, but he's not cut out for a matchup against any of the other seven quarter-finalists United could meet.
Moyes must show his side how quickly it played and how fast it moved the ball because of the Welshman's presence, then find another player to contest the next round who can do that and put in a defensive shift.
If United revert to the slow, monotonous XI they have been recently, they'll go out with a whimper at the hands of a far better side.
Tactical Man of the Match: Ryan Giggs