Athletic saw off Sporting Lisbon 3-1 on the night to reach May's showpiece match in Bucharest, Fernando Llorente's crucial late goal giving the Basques a 3-1 win on the night to take them through 4-3 on aggregate.
They will meet fellow Liga side Atletico, who comfortably held on to a first-leg lead to oust another Spanish side Valencia.
"It's the dream final," said Susaeta, who scored Athletic's opener at San Mames.
"The atmosphere was spectacular and we still haven't grasped what we have done," he added.
"Tomorrow we'll analyse it but we do know we have done something great."
The last time two Spanish teams contested the final of the competition formerly known as the UEFA Cup was in 2007, when Sevilla beat Espanyol on penalties after a 2-2 draw.
Athletic were the only side in the last four never to have won a European trophy, their only previous final coming in 1977 when they lost the UEFA Cup showpiece match to Juventus before most of the current squad were even born.
Marcelo Bielsa's side grabbed the opening goal in Bilbao when Llorente cleverly chested a cross into the path of Markel Susaeta who volleyed past Rui Patricio from close range.
Both sides drove forward at every opportunity and both had chances before Sporting pulled level on the night in the 44th minute when the ball fell to Dutch striker van Wolfswinkel on the edge of the penalty area and he fired low into the corner.
Sporting's aggregate lead lasted a mere two minutes before Llorente got his second assist when he set Ibai Gomez clear to finish with a shot into the far corner in first half added time.
The second period followed a similar pattern to the first, Bilbao's Javi Martinez heading against a post from a corner and Emiliano Insua also striking the frame of the goal with a powerful deflected low drive for Sporting.
With extra time looming, Llorente nipped in front of his marker and diverted a Gomez centre in off the post to send the home fans into raptures.