The visiting side's failure to clear the danger from a corner in the fourth minute cost them the opening goal as Clemens Fritz delivered a cross to find Jannik Vestergaard, who powered home a header to give Bremen the advantage.
Fritz could have punished another example of poor defending after being gifted space on the edge of the box, but he could only shoot straight at Timo Horn.
Bremen were gifted the chance to double their lead before the break when Dominique Heintz foolishly pulled down Anthony Ujah inside the box.
Ujah was unable to capitalise, though, as Horn guessed correctly to keep out the penalty and hand Koln a boost heading into the interval.
Yuya Osako twice threatened to equalise early in the second half, but his efforts were kept out by last-ditch blocks in the six-yard-box as the hosts started to drop deeper to defend.
Felix Wiedwald was suddenly the busier of the two goalkeepers and his save from Svento's powerful strike only briefly relieved the pressure on his back four.
Osako appeared to miss the best chance to secure a point when Wiedwald produced a stunning stop to deny the Japan international after he had found the space to connect with a cross, but Svento reacted quickest in the box to fire home the loose ball with 11 minutes of normal time remaining.