We present the first demonstration of fluorescence correlation spectroscopy (FCS) using superconducting nanowire single-photon detectors (SSPDs) which are free of afterpulses unlike the avalanche photodiode (APD). Multimode-fiber-coupled SSPDs with high system detection efficiency for visible wavelengths were developed and implemented in the FCS system. We performed FCS measurements for Rhodamine B and 6G as fluorescent samples, and found that autocorrelation functions obtained by the SSPDs showed a noise-free curve in the short correlation time region of sub microseconds where the afterpulse effect was dominant using the APD. The obtained results clearly indicate the advantage of SSPDs for the FCS system.