A set \(S\subseteq V(G)\) is an edge metric generator for a graph \(G\) if for any two edges \(e\) and \(f\) of \(G\) there is a vertex \(x\in S\) such that \(d(x, e) \ne d(x, f)\). The minimum cardinality of an edge metric generator for \(G\) is the edge metric dimension \(\mathrm{edim}(G)\) of \(G\). It is proved that if \(G(n,p)\) is the Erdős-Rényi random graph with constant \(p\), then \(\mathrm{edim}(G(n,p)) = (1+o(1))\frac{4\log n}{\log (1/q)}\), where \(q=1-2p(1-p)^2(2-p)\).