A penalty by Sehrou Guirassy just after the hour mark earned the hosts only their second win in 17 Bundesliga and Europa League games this season.
"We've lost a game in which Cologne had one shot on goal other than the penalty," Wenger told reporters after the game at the RheinEnergieStadion.
"We were always missing that little something to get the goal. Cologne defended with spirit and for them it's turned out to be the perfect night.
"Jack Wilshere created goalscoring situations and came close to scoring himself. [Danny] Welbeck, [Olivier] Giroud and Wilshere all looked dangerous, but we were lacking the accuracy and finishing."
The defeat ultimately did not matter for Arsenal in terms of permutations, as the goalless draw between BATE Borisov and Red Star Belgrade in Belarus meant that they qualified as Group H winners regardless.