We report on the possibility of using Silicon Photomultipliers (SiPMs) to detect the scintillation light from neutron conversion in ZnS:⁶LiF scintillators. The light is collected by wavelength-shifting fibers embedded into the scintillator. The difficulty of extracting neutron signals in the presence of high dark count rates of the SiPMs is addressed by applying a dedicated processing algorithm to analyze the temporal distribution of the SiPM pulses. With a single-channel prototype detection unit we demonstrate a very good neutron signal extraction at SiPM dark count rates of about 1 MHz.