Jonas Eidevall’s side carried their impressive domestic form from the Women’s Super League, in which they have yet to drop a point, into the Champions League to show that their 5-0 defeat to Barcelona last month was little more than a blip.
Despite making six changes, Arsenal dominated from the outset and were awarded a penalty in the 15th minute but Nikita Parris missed the opportunity to give the Gunners an early advantage.
Arsenal did not have to wait long though, taking the lead in the 28th minute thanks to a curled free-kick from Stephanie Catley as the Australian lifted it over the wall and in off the bar, out of the reach of Koge goalkeeper Kaylan Marckese.
The Gunners kicked on after the half-time break and Parris made up for her earlier penalty miss just after the hour mark, latching on to a Catley shot to turn and drive home.
Caitlin Foord headed in her side’s third less than 10 minutes later, after Jordan Nobbs’ effort rebounded off the bar.
However against the run of play Koge pulled one back as Arsenal made a number of defensive mistakes before Madalyn Pokorny slotted the ball into the goal.
It would prove to be little more than a consolation goal. Arsenal added two more in the final five minutes to inflict a heavy defeat on the Danish side, with Anna Patten clipping the ball into the top corner before Nobbs rounded off the scoring with a low drive.
Arsenal remain three points adrift of Barcelona, who eased to a 4-0 win over Hoffenheim.
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