The officers had responded about 9 p.m. Tuesday to a report of a man with a handgun in the 1100 block of North Wilmington Boulevard, according to a statement released by the LAPD.
A toy gun was found at the scene by investigators, police said in the statement.
When officers arrived, they saw a lone man in his mid-20s and the deadly encounter unfolded — but authorities provided a vague account of what transpired.
"Believing this may be the suspect from the radio call, they directed their attention toward him; at that time there was an officer-involved shooting," the department said in the statement.
LAPD officials did not say whether officers saw the toy gun before opening fire or whether the man aimed it at police.
Police said several shots were fired and the man fell to the ground. Then the officers' cruiser "traveled forward" and struck the man, pinning him underneath it, police said.
The man was pronounced dead at the scene. The exact cause of his death — gunshot wounds or the weight of the cruiser — has not been determined.
Ed Winter, a deputy chief of investigations for the L.A. County coroner's office, said the man has not yet been identified.
During the deadly encounter, one of the officers suffered an injury to his arm and was taken to Harbor-UCLA Medical Center. Police have not explained how the officer was injured but said he was released Wednesday after treatment.
The incident may have been caught on video as police cruisers in the LAPD's South Bureau are equipped with dash cameras.
The initial police radio call of an "officer down" overstated the extent of the officer's injury, and a dispatcher described the officer as having been shot in the arm.
"Shots fired; can you help," an out-of-breath officer is heard saying into his radio. The officer then advised where to approach the scene from on Wilmington Boulevard.
"Suspect down by vehicle; possible gunfire," the officer said shortly after to a dispatcher. Another officer's voice interjected, "We've got an officer down."
A dispatcher then tells officers over the radio that one of the officers at the scene was shot in the arm, and directs all units to Denni Street and Wilmington Boulevard.
Once the LAPD's specialized shooting investigators arrived, they determined that the officers were not fired at and that the item recovered at the scene was a toy gun.
Initially, a police spokesman had said another suspect was in custody. But as the investigation evolved, LAPD Communications Director Josh Rubenstein said only one suspect — the man who was fatally shot — was involved in the incident.
A multi-agency investigation is underway, which is standard procedure for all LAPD officer-involved shootings. It will be reviewed by the Los Angeles County district attorney's office, the Police Commission and its inspector general.
A white shade tent covered the police car's front end on the sidewalk into early Wednesday morning when coroner's investigators entered the area to remove the man's body.
Tuesday night's shooting is two blocks away from where 17-year-old Fabian Nunez was gunned down earlier that same day.
Times staff writer Joseph Serna contributed to this report.
6:15 p.m.: This article was updated with more information from police about the incident.