Marcelo Bielsa's side have progressed thanks to a 4-3 aggregate scoreline.
In a breathless first half, Bilbao took the lead on 17 minutes when Markel Susaeta volleyed in after Fernando Llorente had cushioned Iker Muniain's pass into his path.
However, the Portuguese side bagged an away goal a minute before the break when Ricky van Wolfswinkel pounced in the box.
Just as Sporting thought that they had gained the advantage, the hosts struck back instantly through Ibai Gomez's chip.
Bilbao came close in the second half when Javi Martinez's header hit the post from a corner. The Spaniards hit the woodwork again minutes later as Emiliano Insua thundered his shot onto the post.
Bielsa's side sealed the win in the final minutes when Llorente met Gomez's low cross.
They will now meet Atletico Madrid in Bucharest on Wednesday, May 9.