Trader Joe's said the man who died was a former employee of theirs.
Police said he was being pursued by officers after allegedly stealing something.
The crash happened as officers attempted to detain him in the roadway, where all three were struck by a vehicle.
Video captured shortly after the collision shows one officer pinned beneath a car, while another officer lay unconscious nearby.
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The suspect was seen lying behind the vehicle and was pronounced dead at the scene.
"I heard yelling and shouting, and then I heard a couple of thuds and a screech of tires. I ran to the window and I saw what had happened," said witness Dennis Roybal.
Roybal said the scene remained visible afterward.
"You can still see the stains on the pavement," he said.
Police said officers were initially responding to a report of a car fire in the Trader Joe's parking lot around 7:30 a.m. when an employee flagged them down about a theft.
The officers then pursued the suspect on foot before the struggle and crash occurred.
Authorities said the driver involved in the crash stopped and is cooperating with the investigation.
"You don't see that too often. Especially right in your neck of the woods and a block away from yourself. It's very scary," said Matthew Calhoun, who lives in the area.
Police remained on scene for hours investigating whether the reported car fire and theft are connected.
Trader Joe's confirmed the suspect was a former employee and said in a statement:
"There was a tragic accident involving a former Trader Joe's Crew Member and members of the San Francisco Police Department that occurred this morning. Our store has been part of this neighborhood for almost 14 years, and we feel compassion for all involved. "
Louis Wong, president of the San Francisco Police Officers Association union, said in a statement that the officers both suffered leg injuries in the collision. "The SFPOA is grateful for the bravery, quick thinking, and selfless actions displayed by these officers under extremely dangerous circumstances. Please keep them and their families in your thoughts as they heal," Wong said.