Aiming to stage an incredible recovery from a 5-1 first leg deficit, Wenger's side briefly looked capable of a famous escape act when Theo Walcott opened the scoring in the first half at the Emirates Stadium.
But any hopes Wenger's recent problems would be erased in historic fashion were brutally dashed by Robert Lewandowski's second half penalty and Laurent Koscielny's dismissal for conceding that spot-kick.
Wenger cut a forlorn figure as Arjen Robben and Douglas Costa netted before Arturo Vidal's double fired Bayern into the quarter-finals with a 10-2 aggregate victory.
It was yet another chastening experience in the Champions League for Wenger, who has now seen Arsenal eliminated in the last 16 for seven successive seasons.
With their Premier League title hopes in tatters and even a top four finish far from guaranteed, Wenger's 184th Champions League match as Arsenal boss may well prove to be his last as speculation grows that he will terminate his 21-year reign at the end of a wretched season.
"It's irresponsible from the referee. It leaves me very angry and very frustrated."
Bayern boss Carlo Ancelotti added: "It was a strange game, honestly the result doesn't reflect what happened on the pitch.
"The penalty took all the pressure out for us."
Arsenal would have to make history to go through as no team in the Champions League era had overturned a four-goal deficit and many Gunners fans gave them no chance judging by the swathes of empty red seats around the Emirates.
Taking possession wide on the right flank, Walcott advanced unchecked by the ponderous Bayern defence and, from an acute angle, he unloaded a powerful strike that flashed over Manuel Neuer's unconvincing attempted save and into the roof of the net.
Walcott had a penalty appeal turned down after Xabi Alonso caught the winger and he was close to a second goal when he surged onto Shkodran Mustafi's pass before firing into the side-netting.
But Giroud, unmarked eight yards out, wastefully headed over from Aaron Ramsey's cross early in the second half, and that would prove Arsenal's final chance.
Lewandowski picked himself up to send David Ospina the wrong way with a clinical spot-kick and the Emirates immediately fell into anarchy.
Faced with an inevitable exit, Wenger retreated to the dug out as insults rained down on him from frustrated fans, and his players waved the white flag as well.
Bayern's second goal in the 68th minute came about after Alexis Sanchez was caught in possession on the edge of his own penalty area, with Robben gleefully picking the Chilean's pocket before slotting past Ospina.
With Arsenal's players just waiting to be put out of their misery, Vidal waltzed through to make it five in the 85th minute and Wenger's agony was complete.
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