Given an exact covering system S = {a_i (mod d_i) : 1 ≤ i ≤ r}, we introduce the corresponding exact covering system digraph (ECSD) G_S = G(d₁n+a₁, ..., d_rn+a_r). The vertices of G_S are the integers and the edges are (n, d_in + a_i) for each n ∈ Z and for each congruence in the covering system. We study the structure of these directed graphs, which have finitely many components, one cycle per component, as well as indegree 1 and outdegree r at each vertex. We also explore the link between ECSDs that have a single component and non-standard digital representations of integers.