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Frenchman Wenger made 11 changes to the team that beat Tottenham Hotspur in last weekend's north London derby, with starts handed to Olivier Giroud, Jack Wilshere and Danny Welbeck.
But the Gunners succumbed to a 62nd-minute penalty from Sehrou Guirassy after the French striker tumbled to ground inside the area under a challenge from Mathieu Debuchy.
Cologne, winless through 12 Bundesliga matches this season, climbed to second with one match to play, as a 0-0 draw between BATE Borisov and Red Star Belgrade in Belarus confirmed Arsenal's status as group winners.
"We've lost a game in which Cologne had one shot on goal other than the penalty," said Wenger. "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."
Seven-time European champions Milan wrapped up first place in Group D with a 5-1 win over Austria Vienna at the San Siro.
Christoph Monschein put the visitors ahead in Italy, but Vincenzo Montella's team roared back with Andre Silva and Patrick Cutrone both netting twice along with a goal from Ricardo Rodriguez.
Atalanta sealed their spot in the knockout phase as Bryan Cristante and Andreas Cornelius notched two goals apiece to thump a listless Everton 5-1 at Goodison Park.
'Disappointing and upsetting'
Robin Gosens added another with a terrific dipping volley, with Sandro Ramirez scoring his first Everton goal as the Toffees crashed to a fourth defeat in five group outings.
"There was nothing riding on the game, but it's our pride. It's not nice to lose games. It's very disappointing and upsetting," Everton captain Wayne Rooney told BT Sport.
Nabil Fekir notched his 13th goal of the season for Lyon in a 4-0 rout of Cypriots Apollon Limassol as the French side joined Atalanta in advancing from Group E.
Mario Balotelli scored twice as Nice moved into the last 32 following a 3-1 victory over Zulte Waregem of Belgium, with Lazio dropping their first points in Group K after a 1-1 draw at home to Vitesse Arnhem.
Marseille snatched a 93rd-minute equaliser courtesy of an own goal from Wilfried Moke in a 1-1 draw at Konyaspor to stay ahead of their Turkish opponents in the race for second behind Salzburg, 3-0 winners over Vitoria Guimaraes, in Group I.
"Miracles don't happen by chance," said Marseille coach Rudi Garcia, whose side saw Jordan Amavi dismissed 10 minutes from time.
"Like in the last match, the team believed in itself until the end despite being a man down."
Villarreal progressed to the knockout stages as Democratic Republic of Congo striker Cedric Bakambu netted a brace to seal a 3-2 victory over Astana in Kazakhstan.
Surprise Swedish side Ostersund -- coached by Englishman Graham Potter -- extended their run in Europe into the New Year with a 2-0 home win over Ukrainians Zorya Luhansk.
Veteran Spain striker Aritz Aduriz scored two penalties in Athletic Bilbao's 3-2 win over Hertha Berlin to leave the Basque club with their fate in their own hands heading into the final round of fixtures.
