“It means a lot to me,” said the 19-year-old Alcaraz. “All the hard work that I put in every day pays off today. To beat Rafa, to beat the best player in history on clay, it means a lot to me.”
Alcaraz had faced 35-year-old fellow Spaniard Nadal twice and lost on both occasions.
His reward is a semi-final with world number one Novak Djokovic, who beat Pole Hubert Hurkacz 6-3, 6-4.
Nadal a five-time winner in Madrid, dropped serve three times in the first set, winning in an eventful match that had to be stopped to assist a spectator in the stands.
Alcaraz had to overcome a sprained right ankle during the match..
In the second set, Alcaraz saved three break points but suffered a fall. He struggled with his right ankle as Nadal won five straight games to take the set.
Alcaraz recovered from the start of the third set, breaking Nadal in the fourth game and then holding serve the rest of the way to win.
