The holders opened the scoring with a brilliantly executed right-foot curler from the outstanding Toni Kroos after 54 minutes that gave substitute keeper Lukasz Fabianski no chance.
Arsenal had earlier spurned an opportunity to go ahead when Mesut Ozil's weak penalty was easily saved by Manuel Neuer in the eighth minute.
The home team were then reduced to 10 men when Szczesny was sent off for clattering Arjen Robben after 38 minutes but David Alaba fired his penalty against the foot of the post.
Midfielder Santi Cazorla was taken off when the Arsenal keeper was dismissed but Fabianski had no chance when Thomas Mueller headed the second goal from a Philipp Lahm cross in the 88th minute.
Asked if Italian referee Nicola Rizzoli was right to send Szczesny off, Arsenal manager Arsene Wenger told Sky Sports television: "If there was contact, the rules are the rules and it is a penalty, but should it have been a red card? I am not sure."
He did not criticise Ozil for his penalty miss after the German tried to dink the ball over Neuer. "Everyone has their own style and that is his," said Wenger.
A year ago Bayern beat Arsenal 3-1 at the same stage of the competition at the Emirates before the Londoners won the second leg 2-0 in Munich to go out on the away goals rule.
"We will have a go in the second leg, you never know. We won 2-0 there last year, we just have to try and do the same again,” said Wenger.


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