Let $R$ be a Noetherian ring, and $x \in R$ be a nonzero divisor. I want to show $$\operatorname{height}\, (x):=\inf\{\operatorname{height}\mathfrak{p}\mid (x) \subseteq \mathfrak{p}\}=1.$$
By Krull's theorem, one only need to show: any prime $\mathfrak{p}$ contains $x$ is not a minimum prime of $R$.
We agree that by Krull's Hauptidealsatz, we already have $height(x)\leq1$, whether $x$ is a zero-divisor or not.
On the other hand, in a noetherian ring $R$ the set of zero-divisors is exactly the union of the prime ideals belonging to the zero ideal $(0)$.
The minimal prime ideals of $R$ are associated to $(0)$ and so if $x$ is not a zero divisor it will be contained in no minimal prime ideal of $R$ and thus $height (x)\geq 1$, which is what you wanted.
The assertion on zero-divisors is true under the sole hypothesis that $(0)$ has a primary decomposition ( automatic in a noetherian ring) and can be found in Atiyah-Macdonald's Commutative Algebra, page 53.