Consider an \(n\times n\) random non-Hermitian matrix \(M\) with eigenvalues \(\lambda_1(M),\cdots,\lambda_n(M)\). The corresponding empirical spectral measure of \(M\) is \(\mu_M=\frac{1}{n}\sum_{i=1}^n\delta_{\lambda_i(M)}\), where \(\delta_x\) is a point mass at \(x\). The author considers the case where \(M\) has a variance profile (i.e., the entries of the matrix are multiplied by some deterministic weights) and a bandwidth \(b_n\) which grows with \(n\) (so that the \((i,j)\)-th entry of \(M\) is zero if \(|i-j|>b_n\) in the aperiodic case, and analogously in the periodic case). The main result of the paper is a central limit theorem for statistics of the form \(n\int f d\mu_M-nf(0)\) for an analytic test function \(f\), under given boundedness and moment conditions. This central limit theorem includes an explicit expression for the asymptotic variance in the setting where \(\lim_{n\to\infty}\frac{2b_n}{n}\leq1\), depending on whether \(M\) is a periodic or aperiodic band matrix.