The shooter, identified by police as Law Thien Huynh, 22, was taken into custody. Police said the shooting was intentional. The weapon was recovered at the scene.
The instructor, 44, died at the scene. Police would not identify the victim until the next of kin was notified.
The Los Angeles Police Department reported the instructor, 44-year-old Robert Herrera, died from his injury.
The shooting happened about 3:15 p.m.
Police interviewed 12 students who were in the class when the shooting happened.
"One of the students left. A 22-year-old Asian student left the classroom, apparently went down to the bathroom, and in a few minutes, came back, either pushed open or kicked open the classroom door and began firing a handgun at the instructor," said LAPD Lt. Paul Vernon.
"The apparent intended victim was the instructor, not any of the other students," said Vernon. "Within two or three minutes of the call coming out, police officers arrived, had the description of the suspect, recognized it was him, and confronted him. While he was initially resistant to give up, wanting to finish smoking his cigarette. In short time they handcuffed him and took him to the police station."
Police did not have a motive for the shooter.
Students in the class described Herrera as a nice person.
"(Huynh) just came in, Robert was at the door, and he just started shooting him," said witness Charles Elliott. "He kept shooting him until the gun was empty."
Coast Career Institute is located at 1354 South Hill St. in downtown Los Angeles.
Lucy Fabian, the girlfriend of the slain teacher, arrived to the school Wednesday night after her phone calls and texts to him went unanswered.
She said the two of them had just come back from Arizona where they told his parents about their plans to get married