We show that dynamical relaxation in the aftermath of a galactic merger, and the ensuing formation of a binary massive black hole (MBH), are dominated by massive perturbers (MPs), such as giant molecular clouds or clusters. MPs accelerate relaxation by orders of magnitude relative to 2-body stellar relaxation alone, and efficiently scatter stars into the binary MBH's orbit. The 3-body star-binary MBH interactions shrink the binary MBH to the point where energy loss from the emission of gravitational waves (GW) leads to rapid coalescence. We take into account the decreased efficiency of the star-binary MBH interaction due to acceleration in the galactic potential, and show that the observed MP abundances in galactic nuclei imply binary MBH coalescence times shorter than the Hubble time. These events are observable by their strong GW emission. MPs thus increase the cosmic rate of these GW events, increase the mass deficit in the stellar core, lead to the ejection of hyper-velocity stars, and suppress the formation of triple MBH systems and the resulting ejection of MBHs into intergalactic space.