Abraham struck a low shot into the bottom corner from outside the area to settle the contest and keep the visitors five points from safety.
The win moved Espanyol up to eighth place, but for their opponents the situation is rapidly becoming desperate. They have not taken a single point since a 1-1 draw with Eibar in mid-January.
Cordoba were the first to threaten, with Aleksandar Pantic volleying over the bar from inside the area following a Federico Cartabia free-kick.
At the other end Lucas Vazquez shot wide of the right post.
Espanyol went ahead in the 43rd minute with a well-worked and superbly-taken goal.
Felipe Caicedo headed a cross out of the area and into the path of Abraham, whose right-footed half-volley from 20 yards arrowed low into the bottom left corner.
Cordoba’s Bebe had a shot saved by Francisco Casilla early in the second half, seconds after Caicedo had worked Mikel Saizar at the other end.
Sergio Garcia de la Fuente might have doubled Espanyol’s lead on the hour mark, but could only shoot wide from six yards.
Cordoba had another chance when Cartabia’s free-kick from the left picked out Florin Andone, but his header was held by Casilla.
But it was the hosts who had the last chance of the game, Victor Alvarez Delgado missing from close range, as Cordoba were left facing up to yet another loss.