Let $Y$ be the support of $\mu$ and suppose that $\mu$ is not ergodic. Then there exists a $T$-invariant measurable set $A\subset Y$ such that $0<\mu(A)<1$. The measures $\mu_1$, $\mu_2$ defined by $$\mu_1(B):=\mu(B \cap A)/\mu(A),$$ $$\mu_2(B):=\mu(B \setminus A)/\mu(X \setminus A)$$ are distinct and invariant, and each gives full measure to the set of points whose orbit is dense in $Y$, contradicting the uniqueness of $\mu$. (Note that in fact the supports of $\mu_1$ and $\mu_2$ are also $Y$, since they each give positive measure to a dense invariant subset of $Y$.)

Let $Y$ be the support of $\mu$ and suppose that $\mu$ is not ergodic. Then there exists a $T$-invariant measurable set $A\subset Y$ such that $0<\mu(A)<1$. The measures $\mu_1$, $\mu_2$ defined by $$\mu_1(B):=\mu(B \cap A)/\mu(A),$$ $$\mu_2(B):=\mu(B \setminus A)/\mu(X \setminus A)$$ are distinct and invariant, and each gives full measure to the set of points whose orbit is dense in $Y$, contradicting the uniqueness of $\mu$. (I haven't checked the details but I think that the supports of $\mu_1$ and $\mu_2$ should also be $Y$, since they each give positive measure to the set of all points with orbit dense in $Y$, and this set is invariant.)