The Dutchman struck twice before the break to extinguish Olympiakos's lead and slotted in a 51st minute free-kick to complete the Reds' 3-2 aggregate comeback against the Greeks.
Here are five talking points...
MOYES SHOULD STILL GO
This was both a false dawn and a stay of execution. United could hardly get any worse after their weekend surrender to Liverpool and, against a side whose first leg victory was owed more to United's pitifulness than the Greek's quality, it was a presentable opportunity for them to lift themselves.
Ryan Giggs rightly urged supporters "not to get carried away" after the win. United should never have allowed themselves to trail such mediocrity, with an Arsenal reserve and Roy Carroll in their squad, by two goals. Moyes deserves credit for the alterations he made which enabled United to qualify for the quarter-finals and he will, briefly, savour his special European night at Old Trafford, but United's withdrawn play following Van Persie's clincher was indicative of their cautious coach.
GIGGS SELECTION PAYS DIVIDENDS
In many ways, it was a typical Ryan Giggs performance. Largely wasteful, often frustrating, yet still pivotal. He effected the game with two key passes that led to Robin van Persie's first-half double and provided United with some of the adventurousness that Michael Carrick and Marouane Fellaini are incapable of. His pass for Van Persie that led to the penalty was reminiscent of his assist for the Dutchman at West Ham last year; that directness will particularly appeal to Moyes. Embarrassingly, the 40-year-old remains United's most creative central midfielder five years on from when Sir Alex Ferguson moved him inside.
Van Persie's third United hat-trick was impeccably timed
OLYMPIAKOS ARE AS BAD AS WE THOUGHT
Jamie Redknapp unwittingly dispelled the myth Olympiakos had suddenly become tricky opponents when he opined they "drew at Benfica", Tottenham's conquerors last week (Spurs are coached by one of Redknapp's friends). United produced their worst away performance for nearly a decade in Athens and were fortunate to be handed such a kind draw back in December. It shouldn't have been this difficult but, as Ferguson used to say, they don't make it easy for themselves.
United v Olympiakos
WELBECK BROUGHT URGENCY
Adnan Januzaj's inevitable drop in performance coincided with the arrival of Juan Mata on the other wing, depriving United of dynamism and alacrity. Welbeck is unlikely to produce a matchwinning performance away from his preferred role up front, but his eagerness and pace unsettled Olympiakos and encouraged Patrice Evra on the overlap. Moyes would have been forgiven for demoting either Robin van Persie or Wayne Rooney, yet he looked elsewhere in an attempt to imbue United with belief and succeeded. Moyes will be acutely aware these are interim solutions, though.
ROONEY FINALLY REPAYS UNITED'S INDULGENCE
There was a moment in the second-half when Wayne Rooney dropped his shoulder to deceive one opponent, jinked past another and burst forward with all the gusto of a teenage Evertonian 10 years ago. He was crafty, committed and indefatigable linking United's attack and revelled from Giggs' presence behind him. There were some merited discussions about whether he should retain his berth in for both club and country, though on this form there is no debate.