Chase-escape is a competitive growth process in which red particles spread to adjacent empty sites according to a rate-$λ$ Poisson process while being chased and consumed by blue particles according to a rate-$1$ Poisson process. Given a growing sequence of finite graphs, the critical rate $λ_c$ is the largest value of $λ$ for which red fails to reach a positive fraction of the vertices with high probability. We provide a conjecturally sharp lower bound and an implicit upper bound on $λ_c$ for supercritical random graphs sampled from the configuration model with independent and identically distributed degrees with finite second moment. We additionally show that the expected number of sites occupied by red undergoes a phase transition and identify the location of this transition.