Producing arguably their best performance of the season, United beat Olympiakos 3-0 to win their pre-quarterfinal tie on 3-2 aggregate. (JPN/Agencies)
Beleaguered 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 English champions overcame a 2-0 first-leg deficit to go through on Wednesday.
Meanwhile, 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.
At the Westfalenstadion, Pierre-Emerick Aubameyang missed a glorious chance to put Dortmund in front before Brazilian forward Hulk put Zenit ahead with a trademark 30-yard piledriver that swerved through the air.
Having lost the first leg 4-2 in Russia, Zenit still needed to score twice more without reply to go through, but Jurgen Klopp's side levelled on Wednesday night when captain Sebastian Kehl headed home a Marcel Schmelzer cross before the interval.
United and Dortmund join reigning champions Bayern Munich, Barcelona, Real Madrid, Atletico Madrid, Paris Saint-Germain and Chelsea in the quarter-final draw, which will be made on Friday at UEFA headquarters in Nyon, Switzerland.
The last-eight ties will be played next month.
Producing arguably their best performance of the season, United beat Olympiakos 3-0 to win their pre-quarterfinal tie on 3-2 aggregate.