Castellanos, on loan from New York City, headed the opener and then ran onto a long ball before slotting past Real's Andriy Lunin to make it 2-0.
Vincius Jr gave Real hope with a header from Marco Asensio's cross.
But Castellanos, 24, tucked in a cross from Yan Couto and headed a fourth - before Lucas Vazquez's consolation.
Real are 11 points behind leaders Barcelona, having played a game more. They play Manchester City in the Champions League semi-final first leg in two weeks.
Castellanos is the first player in over 75 years to score four goals in a La Liga game against Real Madrid. In December 1947, Esteban Echaverria netted five for Real Oviedo against Madrid.
Uncapped Argentine Castellanos had played club football in Chile, Uruguay and the United States before joining Girona on a season-long loan last summer - netting seven La Liga goals before this.
Castellanos only touched the ball 15 times, with as many goals as passes attempted, before being replaced in the 71st minute.
Real were without first-choice goalkeeper Thibaut Courtois through gastroenteritis and striker Karim Benzema with a leg injury.
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