Let $\mathbb{G}_{n,γ}$ be the set of simple and connected graphs on $n$ vertices and with domination number $γ$. The graph with minimum spectral radius among $\mathbb{G}_{n,γ}$ is called the minimizer graph. In this paper, we first prove that the minimizer graph of $\mathbb{G}_{n,γ}$ must be a tree. Moreover, for $γ\in\{1,2,3,\lceil\frac{n}{3}\rceil,\lfloor\frac{n}{2}\rfloor\}$, we characterize all minimizer graphs in $\mathbb{G}_{n,γ}$.