Fellow Serie A sides Atalanta and Lazio also won, while Portugal's Vitoria Guimaraes began their match against Salzburg with not a single European player on the team.
Arsenal come into this game having started the Premier League season poorly after claiming just six points from the opening four games.
Meanwhile, Cologne coach Peter Stoger refused to even discuss the situation at his postmatch news conference.
Cologne fans descended upon Arsenal on Thursday evening chanting the name of former Tottenham defender Kevin Wimmer.
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"That they outsung Arsenal didn't really surprise people though - Arsenal isn't exactly known as a club that creates a particularly tingling atmosphere".
French boss Wenger said finishing in the Premier League's top four was their best route of getting back into the Champions League and so he left out many of his first-team players for the game against Cologne, instead giving those on the fringes an opportunity.
A statement said: "At around 3pm on Thursday, September 14 a large group of football fans gathered in Oxford Street".
Wenger has always maintained his desire to keep Giroud - who has 99 goals for the club - at Arsenal and was full of praise for the striker on the eve of the Cologne clash. They now rank dead last through three games, with just one goal scored and a huge seven surrendered.
However, he wants Arsenal to take the competition step by step and not get too far ahead of themselves as the Europa League remains a hard trophy to win. We struggled to cope with their pressure in the second half. He's an exceptional footballer. We were ready to play at 20:05 BST but these things happen. After the bad first half against Burnley, I think all our performances have been good.
"On the injury front we have no big problems", the Frenchman said.