Van Persie, who left the pitch on a stretcher late on, converted a first-half penalty and scored on the brink of half-time before he struck the final goal of the 3-0 win with a free kick just after the restart. The Theatre of Dreams had 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.
So Moyes, knowing that only a big win would do, told his side to go at Olympiakos from the off, but they were frustrated by the tenacious visitors, who did everything they could to break play up. At least until Giggs lofted a delicious ball over the top towards Van Persie, that is.
Jose Holebas barged over the Dutchman who leapt up to convert the penalty.
David de Gea pulled off a string of saves, keeping his side in the game - the Spanish goalkeeper was at least as valuable as Van Persie, who went on the score two more.
The first of those came when Giggs picked out Rooney. He squared to the Dutchman, who tucked away with the simplest of side-footed finishes.
Five minutes after the restart United got the third goal they needed as Danny Welbeck went down outside the area. Van Persie smacked in the free kick, wrongfooting the goalkeeper and sending his side through.