Before the match, Cologne coach Christoph Daum has called on his players to turn the Mungersdorfer Stadion "into a fortress'', and things appeared to be going to plan thanks to Milivoje Novakovic's first-half strike.
But the hosts failed to add to their lead when Roda Antar missed from the penalty spot, and Eintracht spoilt the party with Fenin popping up nine minutes from time to snatch a point for the visitors.
Eintracht coach Friedhelm Funkel took an injury-depleted squad to Cologne with Christoph Preuss (knee), Aleksander Vasoski (lung surgery), Benjamin Kohler (unfit), Zlatan Bajramovic (toe), Alexander Meier (thigh), Markus Proll (unfit), Umit Korkmatz (broken foot) and Jam Zimmermann (rip) all unavailable for selection.
And the absences appeared to take their toll on the visitors, who failed to produce a single shot on target during the first half.
Cologne were not exactly firing on all cylinders either but eventually found the back of the net after 33 minutes with Umit Ozat providing the pass for Novacovik, who darted through Eintracht's defence before nestling the ball into the right corner from 15 yards.
The goal sparked celebrations among the home supporters, who were further buoyed when a foul by Patrick Ochs on Sergiu Radu saw the referee point to the penalty spot six minutes later.
But Antar was thwarted in his attempts to give Cologne a two-goal lead as visiting goalkeeper Oka Nikolov got a hand to the ball diving to his right.
If the first half had been a disappointing affair, the second half was woeful with neither side able to produce a chance of note until the 82nd minute when Eintracht finally brought a spark of life to the match.
A perfect cross from Ioannis Amanatidis found the head of Fenin, who powered in a header from five yards which left Cologne goalkeeper Farid Mondragon stranded.