Athletic had dominated the early stages and deservedly took the lead when Oscar de Marcos battled his way through the Espanyol defence to fire under Kiko Casilla.
The lead lasted only seven minutes though as Ndri Romaric curled in from outside the area before Espanyol went in front just after half-time thanks to a wonder strike from Vladimir Weiss.
Athletic's threat from set-pieces turned the game around again as Fernando Llorente headed in Markel Susaeta's free-kick before Javi Marquez nodded home seven minutes later to put the hosts 3-2 ahead.
However, the drama did not end there as Albin latched onto Kalu Uche's through-ball in the 93rd minute to power a ferocious shot past Gorka Iraizoz and maintain Espanyol's two-point advantage over Athletic in the race for a Champions League place.
Both sides came into the game in great form with Espanyol unbeaten in their previous six league games and Athletic's only defeat in their last 11 coming to Real Madrid at the Bernabeu.
It was the home side who started the brighter as Llorente headed narrowly wide after just eight minutes.
De Marcos and Iker Muniain also saw efforts fly just off target, but the opener arrived on 26 minutes as De Marcos took advantage of a poor defensive header from Didac Vila to squeeze his shot under Casilla.
The goal seemed to shake Espanyol into life as immediately they created their first chance of the match, but Iraizoz got down well to turn Hector Moreno's effort wide.
There was nothing the keeper could do moments later though as Romaric picked up a loose ball just outside the area and whipped a fine left-footed shot into the far corner.
Marcelo Bielsa introduced Ander Herrera and Susaeta at the break, but Athletic soon found themselves behind as Weiss scored for the second weekend running with a wonderful dipping shot from 30 yards.
Again though the lead did not last long as poor marking from Susaeta's free-kick allowed Llorente the easiest of tasks to head in his sixth goal in three games. And Espanyol did not learn their lesson as minutes later Marquez was given just as much room to flick in Herrera's cross at the back post.
Susaeta then struck the bar as Athletic failed to put the game to bed and that was to prove costly as deep into stoppage time debutant Uche played in Albin and his shot rocketed in off the underside of the bar.