Sokratis Papastathopoulos' goal was initially ruled out by referee Patrick Ittrich for a foul on goalkeeper Timo Horn, but the Video Assistant Referee (VAR) was used to overturn the decision.
Sporting director Jorg Schmadtke said they would demand a replay of the match, prompting Dortmund CEO Hans-Joachim Watzke to brand them sore losers and director Michael Zorc to dismiss the complaint as "ridiculous".
The club announced on Tuesday that they will not launch an appeal and insist they accept Dortmund deserved to win the match.
"The accusation that, in view of the clear result, Cologne do not accept the defeat is wrong," they said in a statement.
"The club never questioned the fact that Borussia Dortmund won the game deservedly, regardless of what we consider to be an incorrect goal.
"Nevertheless, on Tuesday, after a detailed weighing-up of all the factors, Cologne have decided to forgo a formal protest against the awarding of the goal.
"The reason for this, above all, is that the prospects of success for such a protest are low."