The win gave United a 3-2 aggregate win and lifted the pressure on manager David Moyes.
Van Persie struck from the penalty spot in the 25th minute after he had been fouled by Jose Holebas.
And the Dutchman added the second in first-half injury-time with a tidy finish from Wayne Rooney's cross, the second goal coming just five minutes after David De Gea had pulled off a fine double save to deny David Fuster and then Alejandro Dominguez.
Van Persie completed his hat-trick seven minutes into the second half with a free-kick into the left-hand corner.
In Germany, Borussia Dortmund, last season's runners-up, lost 2-1 to Zenit St Petersburg but went through 5-4 on aggregate.
Hulk gave the Russians a 16th-minute lead before Borussia skipper Sebastian Kehl levelled with a 38th-minute header.
Zenit set up a thrilling finish when Jose Rondon slipped his marker in the 73rd minute to head home and make it 2-1 on the night.
Defending champions Bayern Munich, Spanish trio Barcelona, Real Madrid and Atletico Madrid, Chelsea and Paris Saint Germain had already qualified for the last eight.
The draw for the quarter-finals takes place at UEFA's headquarters in Nyon, Switzerland on Friday.
Robin van Persie scored a hat-trick as Manchester United kept their season alive by storming back to beat Olympiakos 3-0 and reach the Champions League quarter-finals on Wednesday.
United were left on the brink of elimination after losing the first leg of the last 16 tie 2-0, but on a stirring night at Old Trafford the Dutchman's treble took them into the last eight for the first time since 2011.
Van Persie's evening was to end on a sour note, as he was stretchered from the field in the latter stages after going down holding his leg, but it did not dampen United's celebrations.
It was the first time United had ever overturned a two-goal deficit in the competition and made them only the sixth team to have clawed back a disadvantage of two goals or more in the Champions League era.
Victory was far from straightforward, however, with United goalkeeper David de Gea notably producing a fine double save shortly before Van Persie levelled the tie in first-half injury time.
United's progress will ease the pressure on manager David Moyes, whose position was the subject of intense press speculation prior to kick-off, but he will remain under scrutiny over the season's closing weeks.
Sunday's 3-0 loss at home to Liverpool left United 12 points outside the top four in the Premier League, meaning that they may have to qualify for next season's Champions League by winning the current edition
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United, for whom Ryan Giggs excelled in central midfield, made their attacking intent immediately clear, with the forwards pressing high up the pitch and sighs echoing around the ground whenever the ball went backwards.
Olympiakos received two early setbacks when Kostas Manolas and Alejandro Dominguez picked up bookings that would have ruled them out of the first leg in the quarter-finals, but they worried United on the counter-attack.
A low Jose Holebas cross had to be hacked clear by Phil Jones, while Hernan Perez should have done better than hoist the ball high and wide from 12 yards after being picked out by Joel Campbell.
Gradually, United raised the tempo, and after Wayne Rooney had put a header against the post from a floated Giggs cross, they secured a foothold in the tie in the 25th minute.
Giggs was the architect, his raking pass freeing Van Persie, who was clattered from behind by Holebas, and after referee Bjorn Kuipers pointed to the spot, the Dutch striker swept home from 12 yards.
Visiting goalkeeper Roberto then touched a Patrice Evra header over the bar, but United would have had a mountain to climb had De Gea not produced a stunning double save to thwart David Fuster and Dominguez.
Within minutes the tie was level, as Giggs — again — freed Rooney with a lofted ball into the inside-right channel and the England man centred for Van Persie to slam home.
United had the momentum and seven minutes into the second half they made it count when Van Persie whipped a 25-yard free-kick past an oddly static Roberto to complete his first Champions League hat-trick.
The hosts remained vulnerable, however, with Olympiakos needing just one goal to regain the lead, and there were heart-in-mouth moments when De Gea was obliged to save from Perez and Dominguez.
Ultimately United survived, but at the cost of Van Persie's injury and a booking to Evra that rules him out of the first leg of their quarter-finals.
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