It was the first time in 29 years that Cologne, who have now snapped their club-record 18-game winless run in the league, have earned three points in Dortmund, with their last win being in 1991.
Dortmund, who had fired five goals past Hertha Berlin last week and won 3-0 against Club Brugge in the Champions League, were lacking any real punch and instead it was hard-working Cologne who scored again thanks to Tunisia international Skhiri with an identical goal from yet another corner on the hour.
Thorgan Hazard cut the deficit with a low shot in the 74th minute but despite desperate late attacks and a stoppage-time golden chance missed by Erling Haaland, who somehow could not push a cut-back over the line from a metre out, they could not get an equaliser.
Dortmund, who have now lost three times in their nine league games, drop to third on 18 points, four off leaders Bayern Munich and two behind RB Leipzig. Cologne's first win of the season lifted them off the bottom and into 15th on six.