For a fixed graph F, the minimum number of edges in an edge-maximal F-free subgraph of G is called the F-saturation number. The asymptotics of the F-saturation number of the Erdos–Renyi random graph G(n, p) for constant \(p\in (0,1)\) was established for any complete graph and any star graph. We obtain the asymptotics of the \(C_m\)-saturation number of G(n, p) for \(m\geqslant 5.\) Also we prove non-trivial linear (in n) lower bounds and upper bounds for the \(C_4\)-saturation number of G(n, p) for some fixed values of p.