 Solution

Let two distinct quotient points correspond to two equivalence classes. By hypothesis those classes lie in disjoint open saturated sets $U,V\subseteq X$. Saturation gives $p^{-1}(p(U))=U$, so the quotient definition makes $p(U)$ open, and likewise $p(V)$.

The image sets contain the two quotient points and are disjoint. Indeed, a class meeting both $U$ and $V$ would be contained in each because both sets are saturated, contradicting their disjointness. Thus the quotient is Hausdorff.