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Two goals in six minutes either side of half-time finally got Hamburg off the mark.
Tottenham loanee Mikey Moore is already winning over Cologne with his ambition to entertain.
Cologne's early lead and a saved penalty slipped away as Füllkrug levelled to earn Bremen a 1-1 draw.
Despite a 4-1 loss, Cologne became the first Bundesliga side ever to field three Englishmen simultaneously.
Bilal El Khannouss dazzled with a solo goal as Stuttgart bounced back in style.