The shooting occurred about 6:15 a.m. near the intersection of Florence and Seventh avenues, a spokesperson for the Los Angeles Police Department said.
The shooter was identified only as a 60-year-old man, according to the LAPD. The male victim was pronounced dead at the scene and the injured woman was transported to a hospital in stable condition.
The deceased man's name was not immediately released.
LAPD homicide detectives were summoned to the scene, where a privacy tent was set up on the sidewalk outside the small business.
The circumstances that prompted the dispute were under investigation.