Hamburg, who needed a relegation playoff to stay up in the past two seasons, have taken just two points from their last six matches. They have lost four of them and are in 12th place on 23 points, only four above the playoff spot.
Their last home win dates back to November.
Cologne, ninth on 26, got their break four minutes before halftime when Hamburg's Johan Djourou tried to dribble past Simon Zoller, who won the ball and fired in without hesitation.
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Hamburg equalised straight after the break, Nicolai Mueller charging through and beating keeper Timo Horn with a stunning left-footed shot from the edge of the box in the 47th.
Cologne tried to find the winner, forcing two superb saves from Rene Adler who tipped Marcel Risse's free-kick onto the bar in the 66th. The visitors also hit the woodwork with an Anthony Modeste header earlier.
Leaders Bayern Munich are eight points clear at the top after drawing 0-0 with Bayer Leverkusen on Saturday with second-placed Borussia Dortmund, on 45, also playing out a goalless draw at Hertha Berlin.
(Reporting by Karolos Grohmann, editing by Alan Baldwin)
