The suspect led police on a chase in a shuttle vehicle before they pulled into the parking lot on Los Feliz Boulevard.
Police placed a spike strip in front of the vehicle, which was parked in a remote corner of the lot.
A view from AIR7 overhead showed big red lettering on the hood of the vehicle, which appeared to say "Toxicology test my corpse and this vehicle."
The person inside the shuttle vehicle was later seen standing in the doorway and engaging with officers.
The standoff did not appear to be affecting operations at Costco or the ability to enter or exit the parking lot.
A bomb squad was called in to make sure the shuttle van was safe. Police say the suspect was wanted for criminal threats.