The radial acceleration relation (RAR) in galaxies describes a tight empirical scaling law between the total acceleration g_tot_(r)=GM_tot_(<r)/r^2^ observed in galaxies and that expected from their baryonic mass g_bar_(r)=GM_bar_(<r)/r^2^, with a characteristic acceleration scale of g_{dagger}_~1.2x10^-10^m/s^2^. Here, we examine if such a correlation exists in galaxy clusters using weak-lensing, strong-lensing, and X-ray data sets available for 20 high-mass clusters targeted by the Cluster Lensing And Supernova survey with Hubble (CLASH). By combining our CLASH data with stellar mass estimates for the brightest cluster galaxies (BCGs) and accounting for the stellar baryonic component in clusters, we determine, for the first time, an RAR on BCG-cluster scales. The resulting RAR is well described by a tight power-law relation, g_tot_{propto}g_bar_^0.51-0.05+0.04^, with lognormal intrinsic scatter of 14.7_-2.8_^+2.9^%. The slope is consistent with the low acceleration limit of the RAR in galaxies, g_tot_=(g_{dagger}_,g_bar_)^0.5^, whereas the intercept implies a much higher acceleration scale of g_{dagger}_=(2.02+/-0.11)x10^-9^m/s^2^, indicating that there is no universal RAR that holds on all scales from galaxies to clusters. We find that the observed RAR in CLASH clusters is consistent with predictions from a semianalytical model developed in the standard {Lambda}CDM framework. Our results also predict the presence of a baryonic Faber-Jackson relation ({sigma}_v^4^{propto}M_bar_) on cluster scales.