The La Liga champions conceded twice in the opening 31 minutes in Piraeus and eventually fell short to start their Group A campaign with a loss.
Simeone lamented his team's profligacy as they were left with work to do in a group that includes Juventus and Malmo.
Kostas Mitroglou restored the hosts' two-goal lead and they held on despite Antoine Griezmann's 86th-minute goal.
"We posed a threat throughout, but Olympiacos were very tough indeed," Simeone said.
"What made the difference was that they were more clinical with the chances that came their way.
"There is no doubt this was a very negative result for us and Olympiacos are now in a very good position. We feel we have a responsibility to bounce back."
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