The two major challenges in Raman spectroscopy are the low intensity of spontaneous Raman scattering and often accompanying luminescence. We overcome these two issues with a novel fiber-dispersive Raman spectrometer utilizing pulsed excitation and a superconducting nanowire single-photon detector (SNSPD). By exploiting chromatic dispersion in the fiber material, we stretched propagation times of Raman photons and performed correlated measurements in the time domain, where the two emission processes, Raman scattering and luminescence, can be effectively separated. The spectrometer greatly benefits from SNSPD metrics, i.e. broad spectral sensitivity (from UV to near-IR wavelength range) on a single-photon level and high timing resolution (small timing jitter), which outperform those of competing avalanche single-photon detectors. The spectral resolution achievable with a fiber-dispersive spectrometer for the optimized components is estimated to be as good as 3 - 10 cm-1 over the Stokes shifted range up to 4400 cm-1 with an excitation wavelength of 785 nm and below 5 cm-1 covering the same range with an excitation wavelength of 532 nm.