David Moyes' side had trailed 2-0 from the first leg in Athens three weeks ago, but produced a stirring comeback with Van Persie, Wayne Rooney, Ryan Giggs and goalkeeper David de Gea returning to the levels of form that led United to the Premier League crown last season.
Van Persie slammed a penalty into the net for the opening goal on 25 minutes before he tucked a Rooney pass beyond visiting goalkeeper Roberto for the second goal a minute into added time in the first period after a lovely ball over the top by Giggs to find Rooney.
The Dutch forward - who was replaced by Marouane Fellaini deep into injury time with what seemed like a dead leg - added number three and the winning goal six minutes into the second half when he lifted a free-kick over the Greek champions' wall and into the net with Roberto left dismally out of position.
Michel's Olympiacos had chances to find the away goal that would have carried them through, but United clung on grimly to secure their place in Friday's draw for the last eight.
After suffering a 3-0 drubbing to bitter foes Liverpool on Sunday, this was an outstanding performance of some character by United that will relieve the pressure on Moyes, who will hope to see his side mount some sort of recovery in the Premier League with his side a lowly seventh, 18 points behind leaders Chelsea.
16' - ROONEY HEADER COMES BACK OFF A POST: Rooney sees a header come back off a post from the Giggs corner before being walloped to safety. Roberto was beaten in the visiting goal. Much, much brighter from United.
38' - DE GEA WITH A DOUBLE STOP: De Gea makes two brilliant saves to prevent United from falling behind. The second one from Dominguez particularly superb as he got back to his feet to make a stunning stop. The first one was from the Fuster header.
25' - VAN PERSIE OPENING GOAL: Holebas goes in late on Van Persie. An awful challenge on the back of RVP. What a chance here for the home side? A gift really. Should never have made that challenge. Van Persie slams the ball to the left of Roberto. The keeper guesses right, but could not keep the ball out of the net. United lead 1-0 on the night.
27' - ROBERTO BRILLIANCE: Roberto with a terrific save to palm a Evra header over the bar from the Van Persie corner. Almost crept in. United almost level at 2-2 on aggregate.
45+1' - VAN PERSIE BRINGS UNITED LEVEL ON AGGREGATE: Rooney plays in Van Persie to slide the ball into the net from six yards. Giggs with the ball into the path of Rooney. Suddenly United lead 2-0. This match well and truly on for the home side.
51' - VAN PERSIE FINDS THE NET FOR A THIRD TIME: The winning goal. Van Persie clips the free-kick over the wall from 25 yards out. Roberto watched it fly past him. Not sure what he was doing. United lead 3-0 on the night.
55' - PEREZ BLOWS GOLDEN CHANCE: Perez sees a free header sail wide of the goal from the Dominguez free-kick. Awful finish. Olympiacos should be 3-1 behind.
78' - MORE DE GEA DEFIANCE: De Gea with a fine stop at his near post from a deflected Dominguez drive before Campbell lifts the ball miles over the bar seconds later. Excellent stuff from De Gea, it must be said.
Manchester United are in their 13th Champions League quarter-final, a total only Bayern can match.
After Barcelona (vs Milan last season), Man Utd are only the 2nd team to lose a Champions League 1st leg 2-0 & progress.
RVP is the fifth Man Utd player to score a Champions League hat-trick after Cole (2), van Nistelrooy, Rooney and Owen.
Robin van Persie is the fourth Dutch player to score a Champions League hat-trick after Rudd van Nistelrooy, Roy Makaay (two each) & Marco van Basten.
It was Ryan Giggs' 140th appearance in the Champions League, just two short of Raul's record.
Greek clubs have conceded at least two goals in 11 of their 12 Champions League games in England.
TALKING POINT: Will this be the turning point for David Moyes? A huge result for United, but perhaps more so for their manager. Claims that he had lost the dressing room and was a dead manager walking were blown apart by these 90 minutes. United were under massive pressure, but all their players responded in some style. It was not perfect, but it was the perfect response to the criticism. It should buy United some breathing space, but they must finish the season strongly in the Premier League. A favourable draw in the Champions League last eight would also be helpful.
MAN OF THE MATCH
Ryan Giggs (Manchester United): A sublime performance by a man who is astonishingly 40. Looked as fresh as a 21-year-old as he pulled the strings from midfield that enabled Robin van Persie and Wayne Rooney to prosper. Makes such a difference to this United side.
Manchester United: De Gea 8, Rafael 7, Jones 7, Ferdinand 7, Evra 7, Valencia 7, Carrick 7, Giggs 9, Welbeck 7, Rooney 8, Van Persie 9. Subs: Fellaini 6, Fletcher 6, Young 6.
Olympiacos: Roberto 5, Maniatis 6, Holebas 7, Manolas 7, Marcano 6, Leandro Salino 6, N'Dinga 5, Perez 6, David Fuster 6,Campbell 6, Dominguez 6. Subs: Machado 6, Valdez 6, Vergos 6.
THE MATCH IN A TWEET
- Sports & Recreation
- Manchester United
- Wayne Rooney
- Robin van Persie
- Ryan Giggs
- David Moyes
- Premier League
- David de Gea