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The suspect is in custody after surrendering to police in the hours-long barricade. All surviving hostages were also released and expected to be OK.
Los Angeles Mayor Eric Garcetti and LAPD Chief Michel Moore said at an afternoon press conference the woman killed was fatally shot, possibly in crossfire, and was pronounced dead at the scene.
Silver Lake Trader Joe's barricade: 1 dead, suspect in custody
Family members of the victim identified her as Melyda Corado, who was an employee at the supermarket. An LAPD source said she was the store manager.
The woman's brother tweeted, "I'm sad to say she didn't make it. My baby sister. My world. I appreciate the retweets and the love. Please respect my family's privacy as we are still coming to terms with this."
Los Angeles police said the incident started with a family dispute at about 1:30 p.m. in the 1600 block of 32nd Street in South Los Angeles, where the suspect shot his grandmother and another female victim, police said.
Relatives said the shooter may have been upset because his 78-year-old grandmother didn't want his girlfriend around the house.
The grandmother suffered about seven gunshot wounds and was listed in critical condition at a nearby hospital, the chief said.
Silver Lake barricade: Police investigate South LA home where crime spree began
The suspect then fled in his grandmother's 2015 Toyota Camry, taking the wounded female victim with him. Police said she suffered a graze wound to her head and is expected to survive. It's believed the woman may be the suspect's girlfriend.
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Officers were able to track the car using LoJack - a stolen vehicle tracking system - and officers tried to stop the car in Hollywood, but the man refused to pull over, Moore said.
During the chase, the suspect fired at officers, shooting out the back window of his car, before he crashed into a pole outside the supermarket, and an officer-involved shooting occurred.
The suspect, who sustained a gunshot wound to his arm, ran into the Trader Joe's, located in the 2700 block of Hyperion Avenue, where several SWAT officers and an armored vehicle surrounded the building.
Frightened customers and workers dove for cover as police bullets fired at the suspect shattered the store's glass doors. Some inside the supermarket climbed out windows as scores of police and firefighters and 18 ambulances converged on the scene and prepared for mass casualties.
Police believe the deceased victim was shot inside the store, possibly during a shootout between officers and the suspect. Officers attempted to render aid to her at the scene, but she died from her injuries.
Shocked customers in the parking lot watched the suspect engage in the gun battle with police.
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"About three or four shots were fired," said witness Makela Wilson. "At that point, they were just telling everyone to get down in their cars and try to duck for safety. Me and a couple of other people were in our cars for a good 30 minutes."
Once inside the supermarket, the suspect took customers and employees hostage "and held them against their will," said Moore.
He added that several law enforcement officers and rescue crews were able to rescue a number of people through other entry points at the store. He said the suspect also released some people himself.
The suspect made a "series of demands" during the standoff but crisis negotiators believed they could convince him to surrender peacefully, Moore said.
"Our hostage negotiators believed they had established a good rapport with him," the chief said.
The hostage situation came to an end through the help of crisis negotiators at about 6:30 p.m., approximately three hours after the ordeal began. The suspect walked out with the rest of the hostages and asked that he put handcuffs on himself.
The unidentified man, who police later said was about 28 years old, was taken into custody without further incident.
Early estimates indicate about 40 to 50 people were inside the supermarket during the incident, including children.
Los Angeles Fire Department officials said they evaluated a total of 10 patients who were not shot. Six people were transported in fair condition and non-life threatening injuries. Those people included the suspect, a 20-year-old woman, 12-year-old boy, 41-year-old woman, 71-year-old woman and an 81-year-old woman.
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A family reunification location for family members affected by the situation was temporarily set up at the LAPD Northeast Station, located at 3353 N. San Fernando Road. You can also call 323-561-3211.
Garcetti congratulated police and firefighters for their work and mourned the loss of life at the Trader Joe's he and his wife regularly shopped at when they lived in the neighborhood.
"The heroism that was shown today was second to none and the teams that were able to respond, secure the perimeter and engage in conversation with the suspect no doubt saved lives today," he said, adding "our hearts go out to everyone who has been traumatized."
Moore later tweeted a tribute to those affected by the incident, saying: "The destructive and tragic consequences one person can inflict are at the forefront of our minds tonight, as is the valor & dedication of our men & women who strived to protect so many innocent people-Our thoughts, prayers & hearts are with all effected by this senseless incident."
Trader Joe's also released a statement on the fatal hostage situation:
"Our hearts are broken over what happened today. This has been an incredible trauma, and our thoughts are with our Crew Members and customers. Our focus is on doing whatever we can to support them at this time. We appreciate everything Los Angeles law enforcement did to ensure an end to this ordeal."
The public is still urged to avoid the area of Hyperion Avenue and Griffith Park Boulevard as an investigation continues through the night and early morning, officials said. The area is not expected to be back open until about noon Sunday.
The Associated Press contributed to this report.