The 25-year-old has been diagnosed with a rupture of the anterior cruciate ligament and a tear of the lateral meniscus in his right knee, the club confirmed.
"Following tests carried out today on our player Rodrygo by the Real Madrid Medical Services, he has been diagnosed with a rupture of the anterior cruciate ligament and a rupture of the lateral meniscus of his right leg," Madrid said in a statement.
Rodrygo, who came on as a second-half substitute and completed the match despite discomfort, later described the setback as "one of the worst days of my life".
"One of the worst days of my life, how much I always feared this injury," he wrote. "A major obstacle has arisen in my life, in my career.
"I'm out of the rest of the season with my club and out of the World Cup with my country, a dream that everyone knows how much it means to me. And all I can do is be strong as always.
"Even though it's a very difficult time, I promise not to stop here."
Recovery from an ACL injury can take several months, with sources indicating Rodrygo could be sidelined for around eight months depending on his rehabilitation.
Brazil, now coached by former Madrid boss Carlo Ancelotti, begin their World Cup campaign in June in the United States, Canada and Mexico.
The Brazilian Football Confederation expressed its solidarity, wishing the winger a speedy recovery.
Rodrygo's injury also prompted a message of support from Neymar, who suffered a similar ACL and meniscus tear in 2023.
"My number 10, my boy, my heir… take care of your head," Neymar wrote. "I am sure you will be back to your best."
Rodrygo made 27 appearances in all competitions for Madrid this season, scoring three goals and registering six assists.
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