Real, who played their fifth match of the current campaign, were beaten 1-0 by newly promoted side Cadiz.
Meanwhile, Barcelona went down 1-0 against Getafe.
Barca are ninth after having played their fourth match.
Here we present the records broken.
Getafe striker Jaime Mata scored a second-half penalty to condemn Barcelona to a 1-0 loss.
Mata slotted his 56th-minute spot-kick into the bottom corner following Frenkie de Jong's foul on Djene Dakonam.
The game saw Lionel Messi strike the post for Barcelona in the first half.
Meanwhile, Anthony Lozano's smart first-half finish gave Getafe their first win against Real in almost 30 years.
Last time Real Madrid and Barcelona lost their respective league games both before the El Clasico was in April 2003 when Los Blancos lost in San Sebastian (2-4) and Barcelona lost at home against Deportivo (2-4).
The two clubs have both lost a La Liga game on the same day for the first time since September 2018.
Barcelona attempted just one shot on target in their 1-0 defeat. Notably, they failed to hit the target in the second half.
Antoine Griezmann registered two shots versus Getafe. He has just scored two goals out of 28 shots in 2020.
Getafe are second in the standings after their third win.
Mata has now scored eight goals from the penalty spot in La Liga.
Real Madrid manager Zinedine Zidane managed his 150th game for Los Blancos in all competitions.
Cadiz have won won a La Liga game against Real Madrid for the fourth time (D3 L18).
This was their first victory since March 1991 (1-0), and it's also the maiden occasion where they notched an away win (D1 L11).
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