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Wayne Rooney struck the post inside the opening 20 minutes as the home side looked to get back into the match, having lost the first leg 2-0 in Athens.
Van Persie was able to get United back on level terms on aggregate just before half time, calmly slotting in from a Rooney cross.
The striker completed his hat-trick shortly after the restart, curling the ball past Roberto Gago with a free kick from just outside the box, leaving the Olympiacos keeper rooted to the spot.
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The visitors almost got one back just a few minutes later as a Dominguez strike deflected off the United goalscorer, but drifted just wide.
De Gea had to remain alert for United, as Holebas unleashed a powerful strike, knowing that an away goal would be valuable for the Greek champions, but he could not get the better of the Spanish keeper.
The visitors continued to press hard as they sought to get back on level terms on aggregate, but De Gea continued to frustrate Dominguez, while Phil Jones turned a dangerous ball into the box behind for a late corner.
Rooney had a chance to put the result beyond doubt late on, but his powerful strike from the edge of the box went just wide.
Van Persie was forced to exit the game early after sustaining a leg injury late on, and had to leave the pitch on the stretcher.
Patrice Evra will miss the first leg of the quarter-final, after receiving a yellow card in the second half.