The Greek side defended a two-goal lead from the first leg, but eventually crashed out of the competition as a Robin van Persie hat-trick meant United won the return 3-0.
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"We had a really good chance, first my header which De Gea saved very well, and then the rebound fell to [Chori] Dominguez and he saved that too," Fuster told the official Uefa website.
"But goalkeepers have a job to do too and he did his very well."
Fuster then went on to stress that he felt proud of Olympiakos' performances in the Champions League, but was unable to hide his disappointment with their elimination at the hands of United.
"Of course, there's a bit of pride – we've had a good campaign and come through a group which was not easy. United are a historic club full of good players and we were close to beating them.
"The team are really disappointed. We had high hopes of going through to the next round, but it wasn't to be. Football is like that – they were able to get three goals and we didn't get any."
The draw for the quarter-finals is scheduled to take place in Nyon on Friday.