The Billy Goats felt Dortmund's second goal of the game, scored by Sokratis Papastathopoulos just before half-time, had been irregularly validated by Video Assist after the referee blew his whistle for a foul which was then overturned by the technology.
Cologne had 48 hours in which they could lodge an appeal against the outcome of the game, but they announced on Tuesday that they will not be taking the matter any further.
*Cologne will not be lodging an appeal against the outcome of the game against Borussia Dortmund
At the same time, they have asked for clarification regarding the incident so as to ensure there will be no repeat in future. "Cologne have always supported the use of video evidence and it would be destined to fail if it did not stick to the conditions that were agreed upon before the start of the season," read a statement on the club's website.
Coach Peter Stöger, speaking ahead of his side's midweek Matchday 5 fixture with Eintracht Frankfurt, said that he feels it is "right that this case has received so much critical attention," adding that "it must also be possible for problem areas to be investigated."
Watch: Dortmund 5-0 Cologne - match highlights
Defeat left the Billy Goats rooted to the bottom of the Bundesliga, still without a point after four matches. Dortmund, meanwhile, lead the pack on ten points with an unbeaten record thanks to goals from Maximilian Philipp (2), Pierre-Emerick Aubameyang (2) and Papastathopoulos at the Signal-Iduna-Park.