The incident happened at the Albertsons market on Wilshire Boulevard.
According to police, a man came up behind the clerk at about 3 p.m. Sunday while she was working at a check-out stand and stabbed her several times around the head and torso. The attacker then ran from the store and drove away.
When police arrived, they found the woman conscious with multiple stab wounds. The 46-year-old woman was transported to an area hospital, where she died Monday morning.
The name of the victim was withheld, pending notification of family.
Police say the woman's estranged husband, 52-year-old Kelvin Green, has been detained as a person of interest.
Anyone with information is urged to call the Santa Monica Police Department at (310) 458-8495.
City News Service contributed to this report.