We investigate theoretically the generation of indistinguishable single photons from a strongly dissipative quantum system placed inside an optical cavity. The degree of indistinguishability of photons emitted by the cavity is calculated as a function of the emitter-cavity coupling strength and the cavity linewidth. For a quantum emitter subject to strong pure dephasing, our calculations reveal that an unconventional regime of high indistinguishability can be reached for moderate emitter-cavity coupling strengths and high-quality factor cavities. In this regime, the broad spectrum of the dissipative quantum system is funneled into the narrow line shape of the cavity. The associated efficiency is found to greatly surpass spectral filtering effects. Our findings open the path towards on-chip scalable indistinguishable- photon-emitting devices operating at room temperature.