The defeat ultimately did not matter in the grand scheme of things for Arsenal, as the draw between BATE and Red Star Belgrade in Belarus meant that they qualified as Group H winners regardless.
But Wenger highlighted that Cologne managed to come out on top despite having just one shot other than the match-winning penalty.
He told reporters: "We've lost a game in which Cologne had one shot on goal other than the penalty.
It's not the way we wanted to do it, but Red Star Belgrade's draw at BATE Borisov means we finish top of our @EuropaLeague group #COLvAFC pic.twitter.com/3Pb1EwUcwp — Arsenal FC (@Arsenal) November 23, 2017
"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 goal-scoring situations and came close to scoring himself. [Danny] Welbeck, [Olivier] Giroud and Wilshere all looked dangerous, but we were lacking the accuracy and finishing."
Nevertheless, Wenger believes that the fact they have already won the group with one match to spare shows that they have accomplished their mission.
"You feel like you've got the job done when you finish top of the group," he added. "We've got BATE on home soil in our final game.
"There's nothing riding on that game [for us], but we want to win it. What it actually means to me to finish top - I can only tell you that after the draw."