Unsettled striker Robin van Persie came off the bench and the Dutchman took the captain's armband from Thomas Vermaelen after replacing Podolski in the 69th minute but got a muted reception from the travelling Arsenal fans.
The Gunners got off to the perfect start in the Rhein Energie Stadion as Santi Cazorla's corner kick was flicked on at the near post by Per Mertesacker for Vermaelen to power in a header at the back post.
Podolski doubled the lead against his former club from the penalty spot in the 15th minute after Alex Oxlade-Chamberlain was brought down in the area.
The German international then got his second three minutes before half-time after a fine pass from Francis Coquelin found Kieran Gibbs on the left. He got to the byline and cut the ball back for Podolski to apply the finish.
Cologne began the second half better but Arsenal's inventive attacking play continued to cause the home side problems and they went further behind in the 60th minute as Gervinho cut in from the wing before slotting in at the near post.