Victory for the visitors would have moved them five points clear at the top of the La Liga table, but Barcelona can now move into first position with a win over Rayo Vallecano on Sunday.
Bilbao, meanwhile, have moved into eighth position with the famous win.
Here, Sports Mole dissects the 90 minutes of action in Bilbao.
On target: 2
On target: 2
Was the result fair?
Even the most ardent of Real Madrid supporters would find it difficult to argue that Bilbao did not deserve all the points at San Mames. There was not too much in the first period, but Bilbao vitally opened the scoring through Aduriz, who found the top corner with a wonderful header. Incredibly, it took Real Madrid until the 58th minute to register their first attempt on target, with Isco striking into the arms of Gorka Iraizoz. The visitors did improve in the second period, but did not create too many opportunities, while Bilbao remained a threat on the counter-attack throughout the 90 minutes.
Late on, as expected, Real Madrid were the team pressing for a goal, but they never looked capable of finding the back of the net, which is perhaps the most worrying part of the defeat. Carlo Ancelotti threw on the likes of Javier Hernandez and Jese Rodriguez in the second period, which brought a change of system, but Bilbao were not too troubled. When Real Madrid lose, the victorious team usually need to have a lot of luck and rely on an excellent performance from their goalkeeper, but that was not really the case here. It was of course a tight match, but Bilbao certainly did enough to deserve all the points.
Having booked their spot in the final of the Copa del Rey on Wednesday, Bilbao would have been forgiven for having some tired legs at San Mames, especially in the second period, but their work rate was quite sensational. Head coach Ernesto Valverde brought in Benat to provide some experience in the middle of the park and the Spaniard, alongside man-of-the-match Mikel Rico, was excellent. The hosts were also without coveted centre-back Aymeric Laporte, but each member of the Bilbao defence was excellent, with Real Madrid simply unable to break through despite their talent.
A special mention must also be given to Inaki Williams, who absolutely ran himself into the ground. The 20-year-old gave the Real Madrid defence a torrid time, with his pace a threat throughout his 75 minutes on the field. Williams proved the perfect foil for Aduriz, who again made the difference for his team. The striker also came close to a second 15 minutes from time, but he rattled the post with a powerful header. The Basque outfit have had a largely difficult season, but a run of just one defeat in seven league games has moved them into eighth position. They have also conceded just twice in their last six league matches - keeping a clean sheet in five of them. Valverde deserves such credit for the upturn in form.
Real Madrid's performance
As mentioned above, it took until the 58th minute for Real Madrid to register their first attempt on target in what was another disappointing display from the league leaders. They dropped home points against Villarreal last weekend, but, having had a free week, were expected to return to winning ways here. Bilbao made much the brighter start, however, and Los Blancos never recovered from the early goal. They did turn the screw for long spells of the second period, especially in the opening 15 minutes, but clear chances remained at a premium. The closest they came in the second half was a Gareth Bale effort from all of 50 yards, which struck the post. To be blunt, they just were not good enough once again.
Perhaps the major problem for Ancelotti is his lack of options on the bench. Luka Modric, James Rodriguez and Sergio Ramos are all still injured and those in reserve - Hernandez, Jese and Lucas Silva - made little impact. It was another off night for Cristiano Ronaldo, while Karim Benzema had one of his poorest games of the season. Bale was perhaps their brightest player in the final third, but the Welshman never really did enough. As it stands, Ancelotti's side are still top of the table, but that might not be the case come tomorrow afternoon. Their form has just dropped off at the worst possible time of the season.
Sports Mole's man of the match
Mikel Rico: There are a number of Bilbao players that could be recognised here, but Rico just gets the nod. The Spaniard set up Aduriz with a super cross for the only goal of the match, while his work in midfield was quite sensational. There was not a moment in the game when Rico was not working, was not attempting to close down the visitors. It was just about the perfect performance from the 30-year-old.
With Real Madrid camped outside the Bilbao box in stoppage time, they needed to work the ball into a wide area and force one last big chance. Ronaldo had provided an option, but Silva decided to take matters into his own hands with a shot and his effort was high and wide of the crossbar, much to the frustration of his teammates. That piece of action summed up Real Madrid's work in the final third - wasteful.
It was a busy afternoon for referee Alberto Undiano. The Spaniard issued seven yellow cards in total, but that was not a particularly high amount when taking into account the number of challenges that occurred. There were a couple of penalty appeals, but Undiano remained in control for long spells and did not really have any major calls to make.
Bilbao: Next up for Bilbao is a trip to Celta Vigo next Saturday.
Real Madrid: Los Blancos, meanwhile, face Schalke 04 in the second leg of their Champions League last-16 clash on Tuesday. Their next league match is at home to Levante on March 15.