A combative start to the game saw three yellow cards in the opening 10 minutes and a nasty black eye for Antonio Valencia.
Robin van Persie had one of his best recent games and has breathed new life into his team's season.
And he has possibly saved his manager's head in the process, after rumours began to spread that Moyes was rapidly losing support in the boardroom.
Speaking to Sky Sports' Geoff Shreeves in his post-match interview, Moyes said:
I'm very happy with the way the players responded after Sunday's defeat.
And the crowd has been incredible. They deserved a good night.
The players were incredibly committed. We didn't play well over there [Greece] but tonight we put it right, the players gave everything.
It's a great competition to be in. We know it's going to be a struggle but we'll give it our best shot.
Talking about his own role, the Scot was relaxed:
I see myself being here for a long time, I've got a big job.
I'm not feeling any pressure from inside the club. I know exactly the job I've got to do, it's probably a bit bigger than I expected when I came in.
I think we know we've got a lot of work to do here.
The Greeks rarely threatened, a magnificent David de Gea double save aside, and that clearly irked Olympiakos manager Michel, who briefly described his thoughts in his post-match interview (via AS, Spanish): "I appreciate the effort of my players. They've done everything possible and for that I am happy. We got in some good positions but didn't take advantage of our chances."
United dominated the first half. Even before Van Persie had opened the scoring from the penalty spot after he was tripped by Jose Holebas, Wayne Rooney had already seen a header come back off the post.
Van Persie was decisive again right on half-time, as a sweeping move from Giggs and Rooney saw the ball moved on to Van Persie, who made no mistake.
It was Rooney and Van Persie who combined again to see the latter bag his hat-trick just five minutes into the second period.
A typical Rooney burst forward saw him shrug off the attention of three Greek defenders. Despite losing the ball, the Englishman won the free-kick, which was dispatched with aplomb.
Immediately, a weight had been lifted and the sense of relief around Old Trafford was palpable.
Displaying a confidence rarely seen this season, United continued the onslaught, but Patrice Evra was just a little too keen and his booking just after the hour mark rules him out for the next game.
As the game wore on into the latter stages, the Greek side came more into it as the Red Devils began to retreat and invite unnecessary pressure.
Clearly irritated, Moyes urged them forward, while Michel looked like he was having a coronary on the touchline.
Some late pressure might have resulted in the away goal that would have crashed the feel-good vibe in Old Trafford.
A late injury to Van Persie resulted in him having to be stretchered from the field, with a chorus of boos greeting his replacement Marouane Fellaini.
It was the only black mark on an otherwise satisfactory night's work for Moyes.
Now, as we await the draw for the quarter-finals, the Scot will ponder what better teams than Olympiakos will arrive in Manchester.
As Champions League qualification through Premier League placings becomes an ever-more distant prize, the manager should perhaps consider making victory in this season's competition his main priority.
Playing with such confidence, Manchester United can still be a force to be reckoned with, and if nothing else, Moyes has gained a few more friends with tonight's qualification.
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