Disney spokeswoman Suzi Brown says the malfunction occurred during a performance Sunday afternoon. No one was hurt.
The park expected to reopen the show, "Aladdin - A Musical Spectacular," sometime Monday.
The musical is based on Disney's popular animated film, "Aladdin." In the movie, Aladdin, with the help of a genie and a flying carpet, woos a young princess.
During Sunday's performance, the carpet flipped over.
A video posted on YouTube shows the two performers, who were strapped in, swaying back and forth above the stage. An announcer apologized for the incident and asked people to leave the theater.