  A graph is well-dominated if all its minimal dominating sets have the same
cardinality. In this paper we prove that at least one factor of every
connected, well-dominated Cartesian product is a complete graph, which then
allows us to give a complete characterization of the connected, well-dominated
Cartesian products if both factors have order at least $2$. In particular, we
show that $G\,\Box\,H$ is well-dominated if and only if $G\,\Box\,H = P_3
\,\Box\,K_3$ or $G\,\Box\,H= K_n \,\Box\,K_n$ for some $n\ge 2$.
