For a digraph $Γ$, a subset $C$ of $V(Γ)$ is a perfect code if $C$ is a dominating set such that every vertex of $Γ$ is dominated by exactly one vertex in $C$. In this paper, we classify strongly connected 2-valent Cayley digraphs on abelian groups admitting a perfect code, and determine completely all perfect codes of such digraphs.