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We reveal the potentially important role of a general mechanism that has hitherto not been invoked in quantum heat management schemes, namely, spectral filtering of the coupling between the heat baths in the setup and the quantum system that controls the heat flow. Such filtering is enabled by interfaces between the system and the baths by means of harmonic-oscillator modes whose resonant frequencies and coupling strengths are used as control parameters of the system-bath coupling spectra. We show that this uniquely quantum-electrodynamic mechanism, here dubbed bath spectral filtering, boosts the performance of a minimal quantum heat manager comprised of two interacting qubits or an analogous optomechanical system, allowing this device to attain either perfect heat diode action or strongly enhanced heat transistor action.