The Spanish club were charged following the home leg of their Champions League semi-final against German opponents Bayern Munich in April.
European governing body Uefa ruled Real must shut two sections of the 81,000-seater stadium in their next tie.
"The fight against racism is a high priority," read a Uefa statement.
"Uefa has a zero-tolerance policy towards racism and discrimination, both on the pitch and in the stands."
The newly-crowned European champions must also put banners bearing Uefa's 'No to Racism' slogan in the closed sections.
The sanction comes after a number of racist incidents involving supporters of Spanish clubs recently.
Last month, Barcelona defender Dani Alves was the target of a racist taunt at Villarreal when a home fan threw a banana on to the pitch next to him.
Levante's Senegal midfielder Papakouly Diop complained of racist abuse from a section of the Atletico Madrid supporters a week later.