Let my write up my comment as an answer. The easiest argument I know uses sheaf cohomology. It requires working through a bit of theory, but ultimately gives a very flexible tool for proving all sorts of things about topological spaces.
Half of this argument could be carried out in singular or de Rham cohomology as well, at least if $Z$ is a CW complex or a submanifold, respectively. But I don't know how to do the comparison between cohomological dimension and covering dimension that way — this ultimately uses Čech cohomology. Also, sheaf cohomology (resp. compactly supported sheaf cohomology) is well-behaved on arbitrary paracompact (resp. locally compact) Hausdorff spaces, so it's exactly suited for this situation where we don't know that much about $Z$.
Lemma. If $X$ is a connected $n$-manifold and $Z \subseteq X$ is closed such that $U = X \setminus Z$ is disconnected, then $\operatorname{covdim} Z \geq n-1$.
Since $X$ and therefore $Z$ is metrisable, I think the covering dimension coincides with the inductive dimension (which I know little about).
Proof. The sequence $0 \to \mathbf Z_U \to \mathbf Z_X \to \mathbf Z_Z \to 0$ of sheaves on $X$ gives a long exact sequence
$$\ldots \to H^{n-1}_c(Z,\mathbf Z) \to H^n_c(U,\mathbf Z) \to H^n_c(X,\mathbf Z);$$
see for instance [Iversen, Ch. III, 7.6]. By Poincaré duality, the latter two are free $\mathbf Z$-modules whose ranks are given by the number of connected components of $U$ and $X$ respectively. The hypothesis $\operatorname{rk} H^n_c(U,\mathbf Z) > \operatorname{rk} H^n_c(X,\mathbf Z)$ therefore forces $H^{n-1}_c(Z,\mathbf Z) \neq 0$.
Since $Z$ is paracompact, it is explained in [Bredon, Ch. II, discussion on p. 122 after Cor. 16.34] that $\operatorname{covdim}(Z)$ agrees with $\dim_{\mathbf Z}(Z)$ (see definition [Bredon, Def. 16.6]). Since $Z$ is paracompact, the preceding proposition 16.5 shows that $\dim_{\mathbf Z}(Z) = \dim_{\mathrm{cld},\mathbf Z}(Z)$ is the 'usual' sheaf-theoretic dimension, i.e. the minimal $k$ such that $H^{k+1}(Z,\mathscr F) = 0$ for all sheaves of $\mathbf Z$-modules $\mathscr F$ on $Z$. Moreover, [Bredon, Cor. 16.33(a)$\Leftrightarrow$(b)] shows that then also $H^{k+1}_c(Z,\mathscr F) = 0$ for all sheaves of $\mathbf Z$-modules $\mathscr F$ on $Z$. Since $H^{n-1}_c(X,\mathbf Z) \neq 0$, we conclude that $\dim_{\mathbf Z}(Z) \geq n-1$. $\square$
References.
[Bredon] G. E. Bredon, Sheaf theory. Graduate Texts in Mathematics 170. Springer, New York, 1997. ZBL0874.55001.
[Iversen] B. Iversen, Cohomology of sheaves. Universitext. Springer, Berlin, 1986. ZBL1272.55001.
