Neil Taylor's wonderful cross from the left was brilliantly headed home by the towering Bony ahead of Thomas Vermaelen. It was the £12million man's 10th goal in his last 16 starts and his 19th goal of the season.
Swansea kept Arsenal at arm's length for the remainder of the first-half, comfortably controlling play with the Gunners' woefully short of ideas.
Santi Cazorla had a chance to level late on, but was denied by Swans keeper Michel Vorm.
Before Podolski's equaliser, however, Michu had a glorious chance to send Swansea 2-0 ahead, only for his header to miss the mark.
On 73 minutes, Podolski scored his first Premier League goal for 2014. Kieran Gibbs' wonderful run down the left was turned home by the German.
Then, just a minute later, Olivier Giroud struck.
Podolski's stunning left-foot delivery was turned into the back of the net by the Frenchman who couldn't miss from six yards out.
Confidence surged through the home team from then on and Wenger even was able to give a debut to Kim Kallstrom in the final ten minutes as Arsenal pushed for a third.
However, Swansea had a kick in it's tail and Mathieu Flamini turned the ball into his own net on 90 minutes.
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