By Chris Harris at Emirates Stadium
Arsenal slipped to the bottom of Group F after a shock home defeat to Olympiacos on Tuesday.
Felipe Pardo struck first for the visitors when his shot flicked off Alex Oxlade-Chamberlain and found the net, but Theo Walcott latched onto Alexis' through-ball to equalise within two minutes.
David Ospina fumbled a corner over his own line to restore Olympiacos' advantage before half-time but Alexis' header at the end of a fine move brought Arsenal level again in the second half.
At that stage the home side looked firm favourites to take three vital points but they were stunned by Alfred Finnbogason's instant response and could not find a third equaliser.
It all means that Arsenal face an uphill task to qualify for the knockout stage. They face Bayern Munich in their next two group games before hosting Dinamo Zagreb and visiting Olympiacos.
SETTING THE SCENE
According to Wenger, Arsenal went into this game with their "backs to the wall" in the Champions League after that surprise defeat to Zagreb in their Group F opener.
That made the visit of Olympiacos as close as you get to a 'must-win' this early in the competition, with a double-header against Bayern Munich looming large. The omens were good - Arsenal had never lost on matchday two and Olympiacos had never picked up a point in England.
Wenger made four changes. Ospina got the nod over Petr Cech, Francis Coquelin returned from injury, Kieran Gibbs replaced Nacho Monreal and Gabriel was back from suspension - so birthday boy Per Mertesacker started on the bench.
Mikel Arteta and Mathieu Flamini missed out through injury while Olivier Giroud served a one-match ban following his red card in Zagreb.
Arsenal started briskly with Hector Bellerin prominent on the right and Alex Oxlade-Chamberlain drifting in off the flank to good effect. But the home side looked most dangerous when they broke at pace after Olympiacos ventured forward.
The first of those counter-attacks was sprung in the 10th minute, Alexis picking out Oxlade-Chamberlain for a shot that found the net on the wrong side of the near post. Walcott was next to race through but Roberto got just enough behind the shot to keep it out.
Olympiacos came into the game more and Kostas Fortounis tested Ospina from the edge of the box. Then Brown Ideye twisted and turned to set up another chance for Fortounis.
The Greek champions were growing in confidence and they took the lead in the 33rd minute. Esteban Cambiasso's corner picked out Pardo on the edge of the box and his effort clipped Oxlade-Chamberlain's heel and found the bottom corner.
Arsenal responded well. Alexis injected some pace on the left flank and cut inside to pick out Walcott's run. This time Roberto could only deflect the shot into his own net.
Normal service seemed to have been restored but there was a twist before half-time. Fortounis curled in a corner, Ospina claimed the ball but then dropped it just over the line.
The climax to the first half set the pattern for the second half - Arsenal pushing forward and probing, Olympiacos looking to soak up pressure and hit on the break.
A glorious touch from Oxlade-Chamberlain briefly threatened to open the door but the home side were kept at bay. They were dealt another blow when Laurent Koscielny was forced off just before the hour mark. On came Mertesacker, and Aaron Ramsey wasn't far behind.
With pretty much his first touch the Welshman played a delicious ball inside the full back for Bellerin. His cross was blocked but the Spaniard had the presence of mind to pick out Mertesacker. His shot was pushed away by Roberto and Cazorla's follow-up was brilliantly cleared by Omar Elabdellaoui.
The pressure was building and Arsenal raised the tempo again, getting their reward with 25 minutes left. A neat, intricate move found Walcott on the right, he clipped in a cross and Alexis powered a header past Roberto.
Again you would have backed the home side to press on. Again they were hit by a sucker-punch. This time Cambiasso lifted the ball over to Pardo on the right and his low cross was turned in by Finnbogason.
Ozil's curling free-kick forced a sprawling save from Roberto as Arsenal looked for a way back into the match for a third time. Then Alexis almost headed in Gibbs' left-wing cross.
Arsenal's efforts became increasingly frantic as time ticked away, but there was no way through.
Referee: Bas Nijhuis
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