The Gunners have now lost their opening two group matches for the first time following a 2-1 defeat at Dinamo Zagreb two weeks ago and are bottom of Group F, six points behind leaders Bayern Munich, who they have to play twice.
Arsenal were shaky at the back and goalkeeper David Ospina's awful mistake gifted Olympiakos a 2-1 lead five minutes before the break when he failed to collect an inswinging corner and allowed the ball to drop over the line.
Alexis Sanchez equalised in the second half but the Gunners seemed to switch off after the goal and Alfred Finnbogason put the Greek champions back in front less than a minute later.
"We lacked discipline. Even when you get a goal back don't rush yourself. We need to get behind the ball and play better defensively," Mertesacker told the club's website (www.arsenal.com).
"Get your discipline back and with possession you get your chances. If we are not disciplined and do not chase the ball we have got no chance.
"We lacked that concentration and it is not understandable why we drifted and did not do the job we are used to doing. That is not acceptable and we will be punished in the Champions League."
The German international said Arsenal had to regroup in time to host the Premier League leaders on Sunday.
"It will take a couple of days to recover and to think about Manchester United and play in a different competition, which will be a good opportunity to come back to the Emirates and play our football with possession and dynamism," he said.
Olympiakos, whose 12 previous visits to England since 1965 had all ended in defeat, scored as many goals in the match as they had managed in the last 50 years on English soil.
"It is not an easy task to come here and get three points," Iceland international Finnbogason said.
"We have had a very, very bad (away) record in Europe up until now and we are very delighted to be rewriting the history of Olympiakos by winning their first game in England," he added.
(Reporting by Shravanth Vijayakumar in Bengaluru; Editing by Peter Rutherford)
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