The first film version of Richard Matheson's zombie apocalypse short story "I Am Legend" came out in 1964 and was called The Last Man on Earth. Luckily it's in the public domain, so you can watch it right here.
Here's the description:
Based on the chilling Richard Matheson science fiction Classic "I am Legend" and later remade as "The Omega Man" starring Charlton Heston and "I Am Legend" with Will Smith. This classic features Vincent Price as scientist Robert Morgan in a post apocalyptic nightmare world. The world has been consumed by a ravenous plague that has transformed humanity into a race of bloodthirsty vampires. Only Morgan proves immune, and becomes the solitary vampire slayer.