We demonstrate resonant coupling of a Mollow triplet sideband to an optical cavity in the strong coupling regime. We show that, in this regime, the resonant sideband is strongly enhanced relative to the detuned sideband. Furthermore, the linewidth of the Mollow sidebands exhibits a highly nonlinear pump power dependence when tuned across the cavity resonance due to strong resonant interactions with the cavity mode. We compare our results to calculations using the effective phonon master equation and show that the nonlinear linewidth behavior is caused by strong coherent interaction with the cavity mode that exists only when the Mollow sideband is near cavity resonance.