The Greek champions held a two-goal lead heading into Wednesday's last 16 second leg clash, but saw that advantage dissipate thanks to a Robin van Persie hat-trick.
Jose Holebas bundled the Dutchman over in the box to give him his first from the penalty spot, before the former Arsenal striker netted a further two either side of the break.
Michel's men saw numerous opportunities to find the net at Old Trafford come and go, leaving the Spaniard to rue a string of missed chances and wondering what might have been.
"I would like to congratulate my players on their effort," he said. "Looking at the result, one might think that it was a one-way match but that wasn't the case.
"We had our chances but failed to convert them."
Olympiacos have now lost on all 12 of their trips to England, conceding an average of three goals.
