Orci was born in Mexico City on July 20, 1973, to a Mexican father and a Cuban mother. His mother immigrated to the United States with her parents after Fidel Castro came to power. Orci grew up in Mexico, and moved with his family to the United States at the age of 10. He was raised in Texas, Los Angeles and Canada. He is the older brother of screenwriter-producer J. R. Orci.
He met his longtime friend and collaborator Alex Kurtzman when both were 17-year-old students at Crossroads, a privately funded school in Santa Monica, California. The first time they came across each other was in a film class, where they discovered each other's love for movies and in particular the Steven Soderbergh film Sex, Lies, and Videotape. The duo found that they had a number of things in common, as Kurtzman had previously lived in Mexico City and the two could relate. Orci later called him an "honorary Hispanic". Orci went on to attend the University of Texas at Austin. The duo got together once again, and began to write scripts. These included one called Misfortune Cookies which Orci described as "loosely autobiographical", and Last Kiss, which Kurtzman said was their version of The Breakfast Club but was set in a lunatic asylum.
The duo modeled their relationship on writers Billy Wilder and I. A. L. Diamond, but were much closer friends rather than just writing partners. They took the approach that they were in a band together and purposely studied teams to discover why they break up. Kurtzman later explained in 2009 that "One big thing that makes the wheels start to wobble is when someone feels that the contribution isn't 50-50. We make sure we live up to the partnership. If we didn't, we wouldn't have lasted this long."