The Gunners were reduced to 10 men just 10 minutes into the crunch game, as Laurent Koscielny saw red for a challenge on Edin Dzeko in the box.
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And, though the Bosnia and Herzegovina international failed to convert the subsequent spot-kick, he later atoned by adding to a James Milner strike to help City to victory.
Podolski, who played 57 minutes of the defeat, later posted on Facebook: "Hey fans, s*** happens. You've seen the match. It was a far cry from what we have imagined.
"But we need to move on, we have many more difficult matches to play. Be with us, we need you! Come on Arsenal! Yours, Poldi."
The 27-year-old has featured in 20 Premier League matches so far this term, scoring on six occasions.