The graphic footage, which appeared on Facebook, shows a violent altercation between a man and several officers in LA's "Skid Row," an area near downtown where many homeless people live.
The video appears to show the man flailing at officers as they try to subdue him.
A voice yelling "Drop the gun!" can be heard before several shots ring out and several onlookers cry out in alarm.
Police had been called to the scene over a reported robbery and were attempting to detain the suspect, who engaged in a fight, Commander Andrew Smith, LAPD spokesman, told AFP.
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During the struggle, police also used a Taser, he said.
The man, whom US media initially reported to be living in one of the area's many tents, was pronounced dead at the scene by paramedics, Smith said, adding that he could not confirm whether the individual was homeless.
Witnesses said the man had been fighting with someone in his tent before the police arrived, the Los Angeles Times reported.
Investigations into the deadly incident will be conducted by an independent inspector general, the district attorney and the LAPD Force Investigation Division, Smith said.
He added that some of the officers are believed to have been wearing body cameras.
Two officers sustained minor injuries, and those who fired on the man have been removed from the field and assigned home for the time being, Smith added.
He added that the coroner would release the individual's identity after his next of kin are notified.
The video comes against a backdrop of highly publicised police shootings of black men that sparked weeks of protests late last year.