Update: 11:10AM Robin van Persie scored a hat-trick as Manchester United beat Olympiakos 3-0 to progress to the Champions League quarter-finals on Wednesday, while Borussia Dortmund joined them in the last eight.
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.
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.
However, a Wayne Rooney header that came back off the post set the tone before the three-time European champions were awarded a penalty midway through the first half when Jose Holebas barged into the back of van Persie.
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 halftime 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.
Evra picked up a yellow card that rules him out of the first leg of their quarter-final while van Persie went off with an injury late on.
"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.
"It's going to be a struggle from here, but we're looking forward to the draw," he added, before saying that Van Persie's leg injury did not appear to be anything serious.
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 the night when captain Sebastian Kehl headed home a Marcel Schmelzer cross before the interval.
Substitute Jose Rondon stooped to head home and win the game for Zenit in the second half.
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 at the beginning of April.