The shooter entered a store in DeKalb County shortly after 1.10pm on Monday, the Georgia Bureau of Investigation (GBI) said.
While checking out, investigators said the suspect got into a dispute with the cashier about his Covid face covering.
He then left the supermarket without paying for his items but returned a short time later with a handgun.
The suspect then shot the cashier. He then began firing at an off-duty DeKalb County Sheriff’s Office (DKSO) deputy who had been attempting to intervene.
Two officers exchanged gunfire with the shooter, and both were wounded, the GBI said.
DeKalb County Police Department officers later arrested a suspect identified as Victor Lee Tucker Jr, 30, of Palmetto, Georgia.
The cashier was taken to Grady Memorial Hospital where she was pronounced dead. Mr Tucker was also taken to Grady and was listed as in a stable condition.
The DKSO deputy is being treated for his wounds at Atlanta Medical Center and is in stable condition. A second cashier was grazed by a bullet. She was treated for her injury at the scene.
The GBI said the DKPD will be taking arrest warrants for Mr Tucker. The GBI also said it will continue its independent investigation.
Once completed, it will be turned over to the DeKalb County District Attorney’s Office for review, the bureau added.