The result gives Sergio Gonzalez's side the upper hand heading into the return clash, which they will host on March 4, and came despite Bilbao's excellent beginning.
Athletic dominated from the off, pressing well and passing excellently, and it was no surprise when Aritz Aduriz gave them an 11th-minute lead.
Aduriz's superb chip - his 15th goal of the season in all competitions - was followed by more Bilbao pressure, but the hosts were unable to double their lead.
And Espanyol took advantage when Victor Sanchez fired them level 10 minutes before the break with a stunning effort from distance.
Espanyol then turned the screw in the second half, with only the crossbar preventing Sergio Garcia from giving them the lead. And the bar came to Bilbao's rescue again when Victor Alvarez nodded into it in the 84th minute.
But despite their missed chances, it was Gonzalez's side who would head home happier after an impressive performance.
The hosts made a bright start, with Aduriz pouncing on a sloppy touch from Anaitz Arbilla before driving towards goal.
And although Aduriz's sixth-minute effort was straight at goalkeeper Pau Lopez, it did not take long for him to open the scoring.
Oscar de Marcos slipped a neat ball between two defenders and Aduriz raced through on goal before coolly lifting over the onrushing Lopez from 15 yards.
It was the finish of a confident man but Espanyol did well from there, soaking up Bilbao's early pressure before Sanchez's wonder strike.
An aimless long ball was poorly dealt with by Bilbao defender Jon Aurtenetxe and Sanchez did not break stride before lashing into the top-left corner from 25 yards.
The nature of Sanchez's strike - which kept moving through the air - gave keeper Iago Herrerin no chance but Bilbao were nearly in front again on the stroke of half-time, with Iker Muniain's curling effort from the edge of the area sailing just wide.
Neither side could be accused of being sluggish after the break, with chances coming thick and fast in the early stages.
First Espanyol's Garcia saw an attempted chip well-read - and saved - by Herrerin before, at the other end, Aduriz shot over from a tight angle.
Unperturbed by his initial effort, Garcia tried another chip, this time from an acute angle near the by-line - and only the woodwork denied him an excellent goal.
Garcia's effort hit the top of the crossbar and Espanyol kept coming, much to the chagrin of the home crowd, as Christian Stuani ballooned Garcia's cut-back over from just inside the penalty box in the 59th minute.
Ecuador international Felipe Caicedo replaced Stuani as Espanyol went in search of a winner and the new man was presented with an excellent chance when, in an offside position, the ball deflected off two Bilbao defenders and played him on.
Caicedo tried to round Herrerin instead of shooting, though, and a heavy touch saw the chance go begging.
Then, with six minutes left, Alvarez was next to try his luck, but unmarked at the back post, he could only head Lucas Vazquez's cross into the bar before the rebound was blasted over, as the visitors had to settle for a point.
The winner of the two-legged tie will face either Barcelona or Villarreal in the final.
