These are apparently the oldest commercially released video games having to do with surviving a zombie attack:

Entombed, for the Atari 2600:

The game cover is rather misleading, though, as it shows an archaeologist being attacked by a ghost, not a zombie:

Zombies for the Sinclair ZX81:

In this case the game cover fits the theme:

This one can be played here with the online ZX81 emulator:

You = X
Zombies = Z
Potholes = 0

Fool the low-IQ zombies into falling into the potholes to win the game.

Both of them released in 1982.

If someone knows of an earlier example of a commercially released zombie-themed video game, let it be known here.