Trailing 4-2 from the quarter-final first leg, Schalke twice took the lead on the night through Klaas Jan Huntelaar and Raul.
But the home side replied with strikes from Ibai Gomez and Markel Susaeta to secure a draw which earned them an aggregate victory and a last-four meeting with Sporting Lisbon.
Bilbao threatened first with an eighth-minute header from Iker Muniain which warmed the palms of visiting goalkeeper Lars Unnerstall.
Schalke responded almost instantly with a testing volley from Huntelaar, which had to be tipped over the bar at full stretch by Gorka Iraizoz in the Bilbao goal.
The German side seemed to take confidence from the effort, and Lewis Holtby also gave Iraizoz something to think about with a long-range left-footed drive which was well held by the goalkeeper.
Huntelaar then put a header over from a few yards out after a corner from Jose Manuel Jurado.
Athletic were struggling to contain their opponents and it was no surprise when they conceded after 28 minutes as in-form Huntelaar notched his 10th goal of the competition, picking up the ball on the edge of the area and drilling a shot into the bottom left corner.
But the Spanish hosts began to find some rhythm as the half wore on and Gomez produced a stunning equaliser four minutes before the break.
The striker, who had come on as a replacement for Andoni Iraola after 30 minutes, raced on to a ball on the left wing, cut across into the centre and smashed an unstoppable shot home from 25 yards out.
Schalke were back in front seven minutes into the second half thanks to Raul, who made the most of a neat lay-off from Obasi to fire home left-footed from the edge of the area.
But Athletic took just three minutes to draw level once more, Muniain supplying the pass for Susaeta to finish into the bottom right corner from 15 yards.
It was an absorbing contest for the San Mames crowd, and Holtby tried to get Schalke back in front with a drive which went narrowly wide of the left post, while Iraizoz needed two attempts to keep hold of a shot from Jefferson Farfan.
Athletic pressed for a winner in the closing stages but Susaeta fired their best chance across the face of goal and past the left post.