Frankfurt had been hoping to go fourth _ the last qualification spot for Europe's top club competition _ but slumped to its first defeat of the year.
The visitors might have had a penalty when Eric Martel struck the ball with his upper arm, before Randal Kolo Muani failed to make the most of a good chance in the 25th. The French forward had scored in each of his previous four games for Frankfurt.
Cologne improved drastically after the break. Timo Hubers scored with a glancing header in the 49th and Skhiri grabbed two more.
Meanwhile, Hertha Berlin defender Marton Dardai scored on his 21st birthday in a morale-boosting 4-1 win over Borussia Monchengladbach to help the team's bid for Bundesliga survival on Sunday.
“That was the best birthday yet,” Dardai said after making his first appearance for Hertha this year.
Hertha had only scored one goal in its previous four games this year - all defeats - but the team came back from Nico Elvedi's early score for Gladbach with goals from Jessic Ngankam, Dardai, Derry Scherhant and Dodi Lukebakio to climb one place above Stuttgart and into the league's relegation playoff place. The bottom two are relegated automatically.
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