Athletic reached their first European final in 35 years when they grabbed a winner two minutes from time to beat Sporting Lisbon 3-1 in their semi-final second leg for a 4-3 aggregate.
Fernando Llorente scored the crucial goal just as the tie was heading for extra-time when he bundled a cross from Ibai Gomex over the line and past Rui Patricio in the Portuguese goal.
In the May 9 final in Bucharest they will face 2010 winners Atletico who beat countrymen Valencia 1-0 for a 5-2 aggregate win.
Just like they had done in the first leg, Athletic went in front against Sporting with a goal after 17 minutes at their San Mames home which took them level on aggregate, but ahead on away goals.
Iker Muniain crossed from the right, the dangerous Llorente chested the ball back, and Markel Susaeta was on hand to volley the ball past Patricio.
The Portuguese were level on the night a minute before half-time when Ricky van Wolfswinkel's left-footed shot beat Gorka Iraizoz in the Athletic goal.
But in time added-on at the end of the first half, Ibai Gomez made it 2-1 on the night to the Spanish side, and 3-3 on aggregate, when he was set up by the impressive Llorente before beating Patricio.
Midway through the second half, both sides hit the woodwork.
Athletic's Javi Martinez crashed a header off the crossbar before former Liverpool player Emiliano Insua sent a drive past Iraizoz but off the post.
As the tie headed towards extra-time, Llorente found the target to put Athletic into a first European final since 1977.
In Valencia, the home side, who had played in the Champions League earlier in the season, mounted a series of attacks against Atletico Madrid in an effort to reduce their 4-2 first leg aggregate deficit.
On the half-hour mark, Spanish international Roberto Soldado got on the end of an Antonio Barragan cross, but his drive just went the wrong side of Thibaut Courtois's goal.
Five minutes before the interval, Sergio Canales shot low and hard but Courtois smothered the danger.
Valencia continued to press after the interval, and they were finally caught on the break on the hour mark as Atletico, the champions in 2010, put the tie to bed.
Adrian Lopez controlled a long pass on his chest and then hit it high into the far corner past Diego Alves in goal.
His strike came just moments after Valencia's Canales had been stretchered off in tears at the Mestalla.
In the 80th minute, Atletico skipper Tiago was sent off for a foul on Soldado and will miss the final.