The gunman, who was not identified, surrendered following a three-hour standoff.
Inside the store, Corado was hit by gunfire, but it was not immediately clear whether the suspect shot the woman, or if she was killed in the crossfire.
"After he was inside the store for a little while they had me crawl away from the scene and run away around the corner", Field said.
Police Chief Michael Moore earlier said the man shot his grandmother seven times and she had undergone a series of surgeries at a local hospital.
"At the termination of the pursuit, an officer involved shooting occured", the police department wrote on Twitter, but did not give further details.
He eventually crashed in front of the Trader Joe's and ran inside. "4-6 people climbed out of a store window". A short time later, a woman was seen being dragged away from the store's entryway and then being treated by paramedics.
A suspect being pursued by police crashed his vehicle, got out and ran into the store on Hyperion Avenue, LAPD Officer Mike Lopez told CNN. Lopez says the suspect shot multiple rounds at officers during the chase but no officers were struck. Police chased the vehicle and exchanged gunfire with the man, who crashed into a pole outside the Trader Joe's in the city's Silver Lake section and ran inside. He crashed the auto attempting to escape police, firing at officers before running into the nearby grocery store.
An employee of The Associated Press who lives in the area reported seeing a crashed vehicle into a utility pole outside the store. Police surrounded the building during the three-hour standoff and several other customers were able to leave the store with their hands up.More news: Dirk Nowitzki reportedly finalizing 1-year pact with Mavericks
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"I regret to inform you that there is one fatality that occurred inside, of a woman", Garcetti said, without further identifying the woman.
Around 8:45 p.m., Albert Corado tweeted that it was his sister, Melyda Corado, who was killed. During the chase, the suspect fired at officers, shooting out the back window of his vehicle. After about 30 minutes, police came inside and rushed some of the customers out, he said.
As the drama unfolded, President Donald Trump tweeted that he was "watching Los Angeles possible hostage situation very closely".
Authorities actively tried to get the barricaded gunman to surrender, as other hostages escaped from windows. Employees snuck out through back emergency exits.
Officials with Trader Joe's issued a statement late Saturday praising police and vowing to help its employees heal.
David Ortiz with the Los Angeles Fire Department confirms more than 100 firefighters responded with 18 ambulances, prepared to transport people to hospitals if necessary.