Cologne were angry about the second goal where they were initially awarded a free kick, only for the referee to change his mind and gave a goal to the hosts after consulting with the VAR.
Cologne, who have no points from their first four league games, announced afterwards that they would lodge a protest and ask for the match to be replayed but on Tuesday said that "the chances of success for such a protest are slim."
The incident happened in the 45th minute at Signal Iduna Park when Cologne goalkeeper Timo Horn dropped the ball at a corner and Dortmund defender Sokratis tapped it into the net.
The Billy Goats repeated their claim on Tuesday that the referee blew his whistle for a foul before the ball crossed the line.
"The ball only crossed the line after the referee's whistle," said the statement. "Therefore, he could not have turned to the VAR and awarded the goal."
The German Football Association (DFB), asked by Reuters to comment on behalf of referee Patrick Ittrich, did not immediately reply.
Both the Bundesliga and Italy's Serie A are trialling the use of VARs this season in co-operation with soccer's rule-making body the International Football Association Board (IFAB).
The IFAB is expected to decide in March whether to permit VARs on a permanent basis. Global soccer body FIFA is hoping to use them at the World Cup in Russia next year.
(Writing by Brian Homewood; Editing by Christian Radnedge)
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