However, the three points were collected in controversial circumstances, with the winning goal having appeared to come off Hannover defender Leon Andreasen's hand.
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Clear goalscoring chances were few and far between during the opening exchanges, with the best of them falling the way of Anthony Modeste, who shot over when well placed.
It was in the 38th minute that the away side struck when from a corner situation, Andreasen diverted the ball into the Koln net with the use of his right hand, but despite protests from many of the home players, the goal stood.
The second half descended into somewhat of a duel between Koln frontman Modeste and Hannover goalkeeper Ron-Robert Zieler, with the latter coming out on top on three separate occasions to keep his team in front.
As the game drew to a close, Koln continued to push forward, but Zieler and the visitors held out to ensure that they have now collected four points from their last two away matches, having won just one of the previous 14 before that.
Consequently, Hannover have jumped up to 14th spot, while Koln have dropped down to sixth.