Did Zariski really define the Zariski topology on the prime spectrum of a ring? The question is not: “Did Zariski really define the Zariski topology?”
It is:  “Did Zariski really define the Zariski topology on the prime spectrum of a ring?”
Here is the motivation. --- On page 80 of
Grothendieck, Alexander, Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : I. Le langage des schémas. Publications Mathématiques de l'IHÉS, 4 (1960), p. 5-228,
availablable here, one reads:

L'introduction de cette topologie en géométrie algébrique est due à Zariski. Aussi est-elle souvent appelée la « topologie de Zariski » de $X$.
The introduction of this topology in algebraic geometry is due to Zariski. This is why it is often called the  “Zariski topology“ of $X$.

The set $X$ is of course the prime spectrum of a ring.
EDIT. Theo Buehler has just posted a question inspired by Qiaochu's answer below. The title of Theo’s question is:
Was Grothendieck familiar with Stone's work on Boolean algebras?

According to Kohls, it was Neal H. McCoy who first introduced the Zariski topology to the prime spectrum in a 1949 paper.
