The Basque outfit, who won the Super Cup in 2021, trailed just after the hour mark when a header from Joao Felix crashed off the post before being deflected into the goal via the back of keeper Unai Simon.
However, the match swung Bilbao’s way inside four minutes late in the second half with both their goals also coming from set-pieces.
A 77th-minute header from Yeray Alvarez levelled the scores before Nico Williams hit the winner nine minutes from time, finding the net from a corner by captain Iker Munian.
The teenage Williams beat Renan Lodi and Atletico skipper Koke to the ball to head past Jan Oblak in goal.
“I’m very emotional right now,” said 19-year-old Nico Williams who lined up for the match alongside older brother Inaki.
“I hope there will be many more joys like this. I’m very happy to have been able to experience this with my Inaki and my mother, who came to hug us at the end of the match.
“She told me I deserved it. The hug with my brother after my goal is a moment I will never forget.”
Oblak admitted his team had been stunned by the defeat.
“It’s hard to explain, we conceded two goals off set-pieces. We have conceded a lot of goals like that this season, and it happened to us again tonight,” said the Slovenian international.
“We have to find the answer to this problem otherwise it will be very complicated until the end of the season.”
Real Madrid made sure of their place in Sunday’s final in Riyadh with a 3-2 win over Barcelona on Wednesday. (AFP)