The incident unfolded inside the store on Laurel Canyon Boulevard around 11:40 p.m. Tuesday, according to the Los Angeles Police Department.
After reviewing security footage, officers determined that a fight broke out between two groups of young men when one of them pulled out a gun and opened fire.
Two of them were struck by the gunfire. Both of them were taken to the hospital, but one did not survive.
That individual was identified as 19-year-old Angelo Cedillo of Granada Hills. The other shooting victim, only described as an 18-year-old man, was said to be in stable condition.
Another 18-year-old, Jorge Armenta, was arrested and booked on suspicion of murder. He was being held on $2,000,000 bail.
There were customers and employees inside the store at the time, but authorities say it was not crowded.
"Any time there's a gun that goes off in a grocery store, I think there's a lot of danger for all the other people around," said Lt. Guy Golan.
Police are now trying to determine what led up to the fight and deadly shooting.