On one of the memorable European nights at Old Trafford, Van Persie scored three times in a 27-minute span either side of halftime to overturn a 2-0 deficit from the first leg. A slew of brilliant saves by David de Gea were also key to a famous win for the English champions — and a vital one for their manager, David Moyes.
Moyes' position had come under intense scrutiny after United's humiliating 3-0 loss to fierce rival Liverpool in the Premier League on Sunday, the latest in a string of dreadful results in his first season in charge.
An early exit in Europe could have stretched the patience of fans and United's hierarchy beyond breaking point — but Moyes is left hoping this can be a watershed moment in his United career.
"I don't want to come out here and say this is the moment," Moyes said, "but at the same time, I really hope it is."
A nerve-shredding night ended with Olympiakos captain Giannis Maniatis on his knees and in floods of tears, and Moyes saluting home fans and wearing the broadest grin.
The under-fire United manager was saved by the goal-scoring qualities of Van Persie, who converted a penalty in the 25th minute, side-footed home his second goal in first-half injury time and curled in a free kick in the 52nd. And he also had words of praise for Ryan Giggs, who at the age of 40 delivered a tireless performance in midfield.
Giggs lasted the full match, despite having played only 81 minutes since Jan. 7, and played a crucial role in Van Persie's first two goals.
"His all-round play was very good," Moyes said. "He defies his age."
Van Persie was carried off on a stretcher in the final minutes after hurting his leg in a challenge but Moyes said he didn't think it was a serious injury.
Olympiakos slumped to its 12th straight away defeat in England and missed out on securing a place in the last eight for the first time since 1999.
The Greek champions had their chances — some of De Gea's saves were breathtaking — but defensively they were chaotic and they became only the sixth team to squander a two-goal lead from the first leg of a Champions League match.
"We were unlucky," Olympiakos midfielder David Fuster said, "and we're pretty upset about it."
In Wednesday's other last-16 match, Borussia Dortmund advanced 5-4 on aggregate despite losing the second leg 2-1 at home to Zenit St. Petersburg. The draw for the quarterfinals is held on Friday.
"We probably now will go into it as underdogs," Moyes said. "If we play to our capabilities, which we have not done too often, then I think we will be a match for any team."
Moyes insisted on the eve of the biggest game of his reign that his position was not under threat, saying United's vision was "long term," the board was still supportive and stressing that he was in a six-year contract.
Fans appeared to be starting to waver, although the banner reading "The Chosen One" — a nickname given to Moyes after he was hand-picked for the job by predecessor Alex Ferguson — remained hung up high in the Stretford End as Olympiakos kicked off.
Initially, United was gripped by panic — misplacing passes, diving into tackles and showing no shape defensively — but the nerves were settled by Van Persie's penalty.
The Netherlands striker tumbled after being barged in the back by Jose Holebas after bringing down a cross-field pass from Giggs, and he got up to convert his penalty unerringly.
Giggs was the architect of the second goal, too, floating another pass to Wayne Rooney, who crossed for Van Persie to caress his finish under goalkeeper Roberto's dive.
The aggregate score was now 2-2 and Moyes greeted the goal with repeated fist-pumps and marched back to the dressing rooms, breaking into a jog at one point.
He was celebrating again within six minutes of the second half when Van Persie curled home a free kick from just outside the area, with Roberto earning the wrath of his defenders for not trusting his wall and vacating the area near where the ball crossed the line.
With Olympiakos attacking relentlessly in the final 25 minutes, the final whistle was met with relief as much as joy by home fans. Languishing in seventh place in the Premier League, the Champions League remains a potential source of silverware for United this season.