Ernesto Valverde's side, fifth, moved within two points of champions Barcelona, fourth, who visit Valencia later on, and third-place Atletico.
Gorka Guruzeta and Nico Williams struck second-half goals to deservedly secure the points for the intense Basques who dominated at a jubilant San Mames.
Atletico have lost four games in La Liga this season, each away from home, while winning all eight matches they have played at their Metropolitano stadium.
"I think we played badly," Atletico goalkeeper Jan Oblak told Movistar.
"We cannot play like this, one way at home, another away, you can't win titles like this, you can't win leagues... Bilbao were a lot better than us."
Athletic drove Diego Simeone's visitors back in the first half, testing Oblak on multiple occasions.
The Slovenian made a superb save low at his near post to deny Guruzeta early on, and another to block Nico Williams' effort after he broke through on the left.
Oblak was saved by the frame of his goal on two occasions, with both brothers Inaki and Nico Williams striking the woodwork.
Athletic's best chance of the first half came from the penalty spot but Oihan Sancet smashed high and wide after Caglar Soyuncu felled Nico Williams.
Athletic got the breakthrough they deserved quickly at the start of the second half, with Ander Herrera crossing to the back post, where Guruzeta was unmarked to finish with ease.
Inaki Williams, making his 400th appearance for Athletic, was instrumental in the build-up and continued his own fine form, even though he did not find the net.
Nico Williams added a stunning second when he made space on the right side of the area and bent a shot over Oblak into the far top corner with his weaker left foot.
Athletic's second goal spurred Atletico into life, with Samuel Lino and Marcos Llorente threatening, but they could not pull the visitors back into the match.
"The fans are giving me everything and I want to keep giving them joy," Nico Williams told Movistar.
"We're an ambitious team, we have to keep working and this is the way... we've taken a step forward from last season and should make our stadium a fortress."
Victory was the perfect way for Athletic to round off their 125th anniversary celebrations.
Basque golfer Jon Rahm took a ceremonial kick-off before the game and showed fans his green jacket, which he won at the 2023 Augusta Masters, while a statue to club legend Jose Angel Iribar was unveiled outside the ground.
Xavi Hernandez's struggling Barcelona could overtake Atletico if they do not lose against Valencia at the Mestalla.
Leaders Girona host Alaves on Monday, while second-place Real Madrid welcome Villarreal on Sunday.
