  We introduce a powerful analytic method to study the statistics of the number
$\mathcal{N}_{\textbf{A}}(\gamma)$ of eigenvalues inside any contour $\gamma
\in \mathbb{C}$ for infinitely large non-Hermitian random matrices ${\textbf
A}$. Our generic approach can be applied to different random matrix ensembles,
even when the analytic expression for the joint distribution of eigenvalues is
not known. We illustrate the method on the adjacency matrices of weighted
random graphs with asymmetric couplings, for which standard random-matrix tools
are inapplicable. The main outcome is an effective theory that determines the
cumulant generating function of $\mathcal{N}_{\textbf{A}}$ via a path integral
along $\gamma$, with the path probability distribution following from the
solution of a self-consistent equation. We derive the expressions for the mean
and the variance of $\mathcal{N}_{\textbf{A}}$ as well as for the rate function
governing rare fluctuations of ${\mathcal{N}}_{\textbf{A}}{(\gamma)}$. All
theoretical results are compared with direct diagonalization of finite random
matrices, exhibiting an excellent agreement.
