Under-pressure United manager David Moyes saw his side produce a much-improved performance from Sunday's 3-0 home defeat to Liverpool, and Van Persie made the difference as the ailing Premier League champions overcame a 2-0 first-leg deficit to go through yesterday.
There was less drama in Germany, where last season's beaten finalists Dortmund lost 2-1 at home to Zenit St Petersburg but progressed 5-4 on aggregate.
United had only previously come back from a first-leg defeat to win a Champions League tie once before, against Roma in the 2007 quarter-finals, and the listless nature of their recent displays would have done little to inspire confidence among the home fans at Old Trafford.
The Dutchman confidently beat Roberto from the spot, and the Spanish goalkeeper then did well to save from a Patrice Evra header.
Olympiakos, who retained the Greek title at the weekend, remained a threat and United goalkeeper David de Gea made a brilliant double save to deny David Fuster and Chori Dominguez as half-time approached.
But United levelled the tie on aggregate in first-half stoppage time when Rooney's low cross was lashed home first-time by Van Persie, and he then scored the decisive third goal, and completed his hat-trick, by curling home a free-kick in the 52nd minute.
"The way the players responded after Sunday's defeat pleased me. I said to them on Monday morning we had to win 3-0 and give the crowd something back," Moyes told British Sky television at full-time.
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