The assembly voted 85 to 18 in favor of the measure, with four abstentions. The requirement already applies to incoming freshmen starting next fall, and extends to transfer students in 2017.
The measure was previously approved by a faculty group and various college committees.
The much-debated requirement has been proposed and rejected three times in the last two decades.
UCLA Chancellor Gene Block has been a proponent of the diversity classes, saying they would help students function in a multicultural society. He said in a statement that the vote confirms UCLA's commitment to exposing undergraduates to diverse backgrounds and views.
"I am delighted that the Academic Senate has approved the diversity-related course requirement," Block said. "This has been one of my long-standing priorities and demonstrates our strong commitment to expose undergraduates to views and backgrounds other than their own."
Opponents said students were already overwhelmed with class requirements, and feared the university would not be able to pay for more classes. Others doubted whether classes could actually improve ethnic relations.
Seven other UC campuses and the UCLA School of Arts and Architecture already have a similar requirement.