Lead-halide perovskites have emerged as promising materials for photovoltaic and optoelectronic applications. Their significantly anharmonic lattice motion, in contrast to conventional harmonic semiconductors, presents a conceptual challenge in understanding the genesis of their exceptional optoelectronic properties. Here we report a strongly temperature dependent luminescence Stokes shift in the electronic spectra of both hybrid and inorganic lead-bromide perovskite single crystals. This behavior stands in stark contrast to that exhibited by more conventional crystalline semiconductors. We correlate the electronic spectra with the anti-Stokes and Stokes Raman vibrational spectra. Dielectric solvation theories, originally developed for excited molecules dissolved in polar liquids, reproduce our experimental observations. Our approach, which invokes a classical Debye-like relaxation process, captures the dielectric response originating from the incipient anharmonicity of the LO phonon at about 20 meV (160 cm-1) in the lead-bromide framework. We reconcile this liquid-like model incorporating thermally-activated dielectric solvation with more standard solid-state theories of the emission Stokes shift in crystalline semiconductors.