After three hours, the suspect surrendered, emerging from the store already handcuffed, with with a group of four hostages, AP said.
Police said Atkins shot his grandmother seven times and a younger woman earlier in the day elsewhere.
Investigators believe the unidentified suspect, in his late teens, shot his grandmother and girlfriend around 1.30 pm local time (2 am Indian Standard Time on Sunday) and then fled in vehicle, AP reported quoting Los Angeles police spokesperson Sergeant Barry Montgomery.
A man who was fleeing from officers crashed his vehicle, a 2015 Toyota Camry, outside of the supermarket and ran inside it, according to USA media.
Officers with riot gear, armed with rifles, stood along the side of the Trader Joe's in the Silver Lake area on Saturday afternoon and used mirrors to try to look inside as hostages periodically came out the front door with the hands raised. During the chase, the suspect fired at officers, shooting out the back window of his vehicle.
"He said he would start killing us if he didn't get to talk to his granny", Lynne Westafer, a Trader Joe's employee told KCAL.
Share this article: Police outside the Trader Joe's where the suspect was barricaded.
Police confirmed there were people still inside the store with the gunman, but they didn't know know how many. Other videos showed an employee dragging a person out of the store and employees escaping on a rope ladder outside a window. Authorities tried to render aid, but she died at the scene.
An LAPD officer told a media briefing the suspect shot at his grandmother and another woman before fleeing in his grandmother's auto with the wounded younger female. Trader Joe's released a statement following the incident.
Frightened customers and workers dove for cover as police bullets fired at the man shattered the store's glass doors.
Police said they used the car's LoJack system to track the vehicle, and then got into a gunfight with the man when they tried to pull the auto over.
The suspect's girlfriend was taken to a hospital in fair condition. She was initially taken to a hospital in grave condition, and after surgery was upgraded to critical condition, the police chief said.
Officials said about 40 store employees and customers were inside.
Melyda Corado, 27, manager of the Silverlake Trader Joe's was identified as by her brother Albert Corado as the woman who was killed. The suspect, who has not been identified, is said to be about 28 years old. It's unclear whether she was hit by officers or the suspect.
He then saw about five employees hang out a second-floor window and drop to the ground, and about 15 other people run to safety from the back of the store.