Willian Jose put the Basques ahead in the third minute and after Madrid missed a series of clear chances, had two penalty appeals waved away and saw Lucas Vazquez sent off, midfielder Ruben Pardo sealed their fate with a headed goal in the 84th.
Many home fans flocked away early in dismay while others greeted the final whistle with deafening boos after Real Sociedad, in their first game under new coach Imanol Alguacil following Asier Garitano's sacking, won at the Bernabeu for the first time since 2004.
The defeat left Madrid, who drew 2-2 with struggling Villarreal on Thursday, fifth on 30 points after 18 games in the La Liga standings, 10 adrift of Barcelona who won 2-1 at Getafe.
Real Sociedad climbed from 16th up to 11th.
Before kick-off, Madrid paraded the Club World Cup trophy they won last month for a third consecutive time but the celebratory mood was soon punctured when Real midfielder Casemiro was penalised for hauling down Mikel Merino in the box.
Jose Willian smashed his spot-kick down the middle of the goal beyond Thibaut Courtois and although Real rallied with a glut of chances throughout they somehow failed to find the net.
Karim Benzema missed a gilt-edged opportunity by firing wide of the far post with the goal gaping, while Vazquez struck the other post with what looked like an intended cross.
Real spurned a triple chance immediately after the interval as Sociedad goalkeeper Geronimo Rulli thwarted Vinicius Jr. and Raphael Varane from close range, and their task was made all the harder when Vazquez was dismissed in the 61st minute for a second booking.
Argentine keeper Rulli later appeared to trip Brazilian teenager Vinicius in the area but no penalty was awarded, a decision which left the Bernabeu faithful incensed after Madrid captain Sergio Ramos had an earlier penalty shout turned down.
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Barcelona's Lionel Messi shoots at goal - La Liga Santander - Getafe v FC Barcelona - Coliseum Alfonso Perez, Getafe, Spain - Janaury 6, 2019. (Photo: Reuters)
Lionel Messi scored for the seventh time in four La Liga games as champions Barcelona scrapped their way to a hard-fought 2-1 win at Getafe that propelled them five points clear at the top of the table.
Messi, La Liga's top scorer this season, opened his account for 2019 with his 16th goal, giving Barca a 20th minute lead as he tucked the ball into the net from a tight angle after his initial shot was saved by the legs of Getafe keeper David Soria.
It took the Argentine's career tally to a staggering 399 goals in La Liga before Uruguay striker Luis Suarez doubled Barca's advantage in the 39th with a spectacular volley from outside the area.
Yet Getafe hit back two minutes before the interval with a close-range finish from striker Jaime Mata after a splendid team move and continued to push hard after the break.
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Atletico Madrid's Antoine Griezmann scores their first goal past Sevilla's Tomas Vaclik Soccer Football - La Liga Santander - Sevilla v Atletico Madrid - Ramon Sanchez Pizjuan, Seville, Spain - Janaury 6, 2019. (Photo: Reuters)
Earlier, a stunning free kick goal from Antoine Griezmann earned Atletico Madrid a 1-1 draw against Sevilla in an entertaining game between La Liga's second and third-placed sides.
Sevilla took a deserved lead at a packed and rocking Sanchez Pizjuan stadium in the 37th minute with a close-range strike from Wissam Ben Yedder following a clever team move, shortly after striker Andre Silva had smacked the post for them.
Ben Yedder missed a clear chance to double Sevilla's advantage but was thwarted by Atletico goalkeeper Jan Oblak, before Atletico drew level at the end of the first half through Griezmann's sizzling set-piece.
The draw left Atletico second in the standings on 35 points with Sevilla third on 33, while leaders Barcelona moved on to 40 points after beating Getafe 2-1, stretching their advantage over Atletico to five points after 18 games.
