For a commutative ring $R$ with identity, the ideal-based zero- divisor graph, denoted by $\Gamma_I(R)$, is the graph whose vertices are $\{x\in R \backslash I| xy \in I$ for some $y \in R \backslash I\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy \in I$. We investigate an annihilator ideal-based zero-divisor graph, denoted by $\Gamma_{Ann(M)}(R)$, by replacing the ideal $I$ with the annihilator ideal $A_{nn}(M)$ for an $R$-module $M$. We also study the relationship between the diameter of $\Gamma_{Ann(M)}(R)$ and the minimal prime ideals of $A_{nn}(M)$. In addition, we determine when $\Gamma_{Ann(M)}(R)$ is complete. In particular, we prove that for a reduced $R$-module $M$, $\Gamma_{Ann(M)}(R)$ is a complete graph if and only if $R \cong \mathbb{Z}_2 \times \mathbb{Z}_2$ and $M\cong M_1 \times M_2$ for $M_1$ and $M_2$ nonzero $\mathbb{Z}_2$-modules.