We study the statistics of Dirichlet eigenvalues of the random Schrödinger operator $-\epsilon^{-2}\Delta^{(\text{\rm d}\mkern0.5mu)}+\xi^{(\epsilon)}(x)$, with $\Delta^{(\text{\rm d}\mkern0.5mu)}$ the discrete Laplacian on ${\Bbb Z}^d$ and $\xi^{(\epsilon)}(x)$ uniformly bounded independent random variables, on sets of the form $D_\epsilon:=\{x\in{\Bbb Z}^d\colon x\epsilon\in D\}$ for $D\subset{\Bbb R}^d$ bounded, open, and with a smooth boundary. If ${\Bbb E}\xi^{(\epsilon)}(x)=U(x\epsilon)$ holds for some bounded and continuous $U\colon D\to{\Bbb R}$, we show that, as $\epsilon\downarrow0$, the $k$th eigenvalue converges to the $k$th Dirichlet eigenvalue of the homogenized operator $-\Delta+U(x)$, where $\Delta$ is the continuum Dirichlet Laplacian on $D$. Assuming further that $\text{\rm Var}(\xi^{(\epsilon)}(x))=V(x\epsilon)$ for some positive and continuous $V\colon D\to{\Bbb R}$, we establish a multivariate central limit theorem for simple eigenvalues centered by their expectation. The limiting covariance for a given pair of simple eigenvalues is expressed as an integral of $V$ against the product of squares of the corresponding eigenfunctions of $-\Delta+U(x)$.