Schalke threatened early on and took a deserved lead through the prolific Klaas-Jan Huntelaar, who drilled a classy opener in from 20 yards.
But Bilbao pinned them back before the break courtesy of a sumptuous strike by substitute Ibai Gomez.
Schalke edged ahead through Raul's emphatic strike, but Markel Susaeta put the game out of reach with a low finish. The result was soured only by the yellow card for Javi Martinez, which rules him out of the first leg of the semi-final against Sporting Lisbon.
Having been ambushed 4-2 at the Veltins-Arena last week, Schalke found themselves needing to win by three clear goals at the San Mames. Quite the task given how impressive Marcelo Bielsa's side have been in Europe's second tier club competition this season.
And Schalke were forced to repel some early pressure from the Basque side, who came close when Iker Muniain found the hands of Lars Unnerstall with a header after his run was picked out by Oscar De Marcos's beautiful lofted pass over the defence.
But Huntelaar has been red-hot this season and he looked in the mood as he lashed a volley just over the bar. The Dutchman was a constant menace and put Schalke ahead with a fine low effort from 20 yards.
Rather than kick-start a full Schalke revival, however, the German side were pinned back as Athletic Bilbao reacted with wave after wave of pulsating attacks with Muniain and then Fernando Llorente coming close for the home side.
Bilbao did not relent and Gomez, a first half substitute for the injured Ander Herrera, raised the roof at the San Mames, unleashing a majestic curling effort from 20 yards that nestled in the bottom corner.
The goal put Schalke back to square one, and seemed to have terminally damaged their hopes of an unlikely fightback.
But hope was renewed when the Bundesliga side edged ahead again on the night courtesy of an emphatic strike from Raul, who notched his 77th goal in European competition.
Running onto Chinedu Obasi's lay-off on the edge of the box, the former Real Madrid striker lashed into the top corner with a shot that Gorka Iraizoz could get nowhere near. But just as Schalke have done throughout this two-legged affair, they threw their lead away soon after.
Collecting Muniain's perfectly-weighted through ball, Susaeta squeezed a low shot inside the near post to restore parity on the night.
Schalke continued to threaten through Obasi and substitute Jefferson Farfan, but could not find the goal to make Bilbao sweat in the final moments.
Bilbao, who famously knocked out Manchester United in this competition, now face Manchester City's conquerors in what promises to be another thrilling clash.