Deasy has served as superintendent to three other districts around the country, and he's been credited with raising test scores at Santa Monica High School and its feeder schools.
Cortines, 78, had previously announced that he would likely be stepping down on April 1, although his contract runs through December.
A.J. Duffy, president of United Teachers Los Angeles (UTLA), said the board should have held a national search, and many felt there would be other candidates before someone was selected.
"We're the nation's second-largest school district. That warrants a national search," said Duffy. "UTLA should have been part of the process."
The district never announced a search for a replacement.
Cortines said the district was only trying to be fiscally responsible. The district didn't have to spend any money recruiting candidates.
Deasy has impressed reformers at the Los Angeles Unified School District with his intensity about turning around underperforming schools.
Doc Ervin, assistant superintendent at the Partnership for Los Angeles Schools, which runs 15 LAUSD campuses, said he found Deasy scholarly and hands-on after meeting with him several times during informal mentoring sessions.
"Man, this guy is really serious about education," Ervin said. "It's clear he's in charge, but he also wants to hear from people."
The Associated Press contributed to this story.