We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If $1/n \ll p = 1-Ω(1)$, then, with high probability, there is a set of $n - o(n)$ vertices that is partitioned in the same manner by all maximum cuts of $G_{n,p}$. Moreover, the analogous statement remains true when one replaces maximum cuts with nearly-maximum cuts. We then demonstrate how one can use this statement as a tool for showing that certain properties of $G_{n,p}$ that hold in a fixed balanced cut hold simultaneously in all maximum cuts. We provide two example applications of this tool. First, we prove that maximum cuts in $G_{n,p}$ typically partition the neighbourhood of every vertex into nearly equal parts; this resolves a conjecture of DeMarco and Kahn for all but a narrow range of densities $p$. Second, for all edge-critical, nonbipartite, and strictly 2-balanced graphs $H$, we prove a lower bound on the threshold density $p$ above which every largest $H$-free subgraph of $G_{n,p}$ is $(χ(H)-1)$-partite. Our lower bound exactly matches the upper bound on this threshold recently obtained by the first two authors.