The incident happened Tuesday morning, according to the Los Angeles Police Department. The suspect, identified as 32-year-old David Sumlin, was arguing with the dog's owner when he put a pillow over the dog.
He apparently threatened the owner before shooting the dog through the pillow twice.
Responding officers took the puppy to a nearby animal shelter where it died.
Additional details about the relation between the dog's owner and the suspect were not available.
Sumlin is now wanted for domestic violence and animal cruelty. According to authorities, he often frequents the area of 7th Street and Stanford Avenue in downtown L.A.