Marco Reus, who was initially named on the bench before coming in as a late replacement for the injured Erik Durm, came close to breaking the deadlock early on with a low effort which drifted just past the post.
The hosts struggled to interrupt Dortmund's dominance, but chances proved to be rare after the early opportunity for Reus as the Koln defence held firm for long spells.
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A rare venture forward from Koln could have produced the opening goal against the run of play had Matthias Lehmann managed to keep his effort from flying into the stands from the edge of the box.
Reus was proving to be a crucial figure in the final third of the pitch for Dortmund, and he came close to breaking the deadlock again when Timo Horn was forced off his line to make a block.
It was the hosts who took the lead five minutes before the interval when Marcel Risse picked out Vogt, who finished emphatically from the edge of the box.