Sergio Garcia opened the scoring against the run of play for the visitors in the 29th minute after a frantic goalmouth scramble, before Christian Stuani wrapped up the win on the hour mark with a fine individual effort.
Defeat means Eibar remain a point and a place above the drop zone with two games left, but they have now lost 15 of their last 18 matches since they were in eighth place on January 10.
Eibar flew out of the blocks in determined mood, seizing control of affairs from the off, and they had the first sight on goal as early as the third minute.
Saul let fly from just outside the box but Espanyol keeper Kiko Casilla showed great reflexes to adjust well and tip it over the bar.
The hosts continued to dominate possession but found chances hard to come by and Espanyol made them pay by taking the lead just before the half-hour mark.
Stuani’s header forced a superb save out of Eibar keeper Xabi Irureta at full stretch diving to his left, but the hosts were unfortunate to see it fall directly to Felipe Caicedo.
The Ecuadorian’s snap-shot ricocheted off the near post across the box, creating panic in Eibar’s ranks, and the visitors once again got the rub of the green as the ball fell to Garcia.
The former Barcelona striker showed great composure amid the chaos to rifle into the roof of the net from an acute angle.
It could have got even worse for the strugglers as Espanyol probed for a second before the interval, but Irureta pulled off a string of saves to keep it at 1-0.
His opposite number Casilla was lucky to stay on the pitch five minutes into the second half.
With Eibar looking for an early equaliser, the Espanyol custodian raced off his line and clumsily brought down substitute Dejan Lekic just outside the area.
The only reason the card shown was yellow and not red was that, luckily for Casilla, there was a defender tracking back so he was not necessarily last man.
Uruguayan international Stuani then took centre stage in the 60th minute, juggling the ball from one foot to the other and rifling a left-footed volley through Irureta’s legs from close range to secure the win, which lifts Espanyol to seventh.